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Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk-Ulam Theorem and Hilbert’sNullstellensatz— and vice versa
Dilip P. Patil
Department of MathematicsIndian Institute of Science, Bangalore
Bhaskaracharya Pratishthana1, PuneJuly 22, 2018
1On the occasion of Birthday of Late Professor Shreeram S. Abhyankar(1930 – 2012), Founder Director, Bhaskaracharya Pratishthana, Pune.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Shreeram Shankar Abhayankar
Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012)Founder Director
Bhaskaracharya Pratishthana, Pune.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Shreeram Shankar Abhayankar
Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012)
Founder DirectorBhaskaracharya Pratishthana, Pune.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Shreeram Shankar Abhayankar
Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012)Founder Director
Bhaskaracharya Pratishthana, Pune.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Abstract
The main aim of this talk is to provide an algebraic proof of thewell-known Borsuk-Ulam theorem by using projective algebraicsets. In fact, we prove a more general form of Borsuk-Ulamtheorem called the Borsuk-Ulam’s Nullstellensatz by establishingits equivalence with the real algebraic Nullstellensatz.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Uwe Storch
Discussion and Ideas of Proofs are based on :
The personal discussions with late Prof. Dr. Uwe Storch(1940 – 2017), Ruhr-Universität Bochum, Germany and his lectureon 23 January 2003, on the occasion of 141-th birthday of Hilbertat the Ruhr-Universität Bochum, Germany.
Uwe Storch (12 July, 1940 – 17 Sept, 2017)
The recent preprint (jointly with Kriti Goel and Jugal Verma)Nullstellensätze and Applications, IIT Bombay, 2018.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Uwe Storch
Discussion and Ideas of Proofs are based on :
The personal discussions with late Prof. Dr. Uwe Storch(1940 – 2017), Ruhr-Universität Bochum, Germany and his lectureon 23 January 2003, on the occasion of 141-th birthday of Hilbertat the Ruhr-Universität Bochum, Germany.
Uwe Storch (12 July, 1940 – 17 Sept, 2017)
The recent preprint (jointly with Kriti Goel and Jugal Verma)Nullstellensätze and Applications, IIT Bombay, 2018.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Uwe Storch
Discussion and Ideas of Proofs are based on :
The personal discussions with late Prof. Dr. Uwe Storch(1940 – 2017), Ruhr-Universität Bochum, Germany and his lectureon 23 January 2003, on the occasion of 141-th birthday of Hilbertat the Ruhr-Universität Bochum, Germany.
Uwe Storch (12 July, 1940 – 17 Sept, 2017)
The recent preprint (jointly with Kriti Goel and Jugal Verma)Nullstellensätze and Applications, IIT Bombay, 2018.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Uwe Storch
Discussion and Ideas of Proofs are based on :
The personal discussions with late Prof. Dr. Uwe Storch(1940 – 2017), Ruhr-Universität Bochum, Germany and his lectureon 23 January 2003, on the occasion of 141-th birthday of Hilbertat the Ruhr-Universität Bochum, Germany.
Uwe Storch (12 July, 1940 – 17 Sept, 2017)
The recent preprint (jointly with Kriti Goel and Jugal Verma)Nullstellensätze and Applications, IIT Bombay, 2018.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Late Professor Uwe Storch
Discussion and Ideas of Proofs are based on :
The personal discussions with late Prof. Dr. Uwe Storch(1940 – 2017), Ruhr-Universität Bochum, Germany and his lectureon 23 January 2003, on the occasion of 141-th birthday of Hilbertat the Ruhr-Universität Bochum, Germany.
Uwe Storch (12 July, 1940 – 17 Sept, 2017)
The recent preprint (jointly with Kriti Goel and Jugal Verma)Nullstellensätze and Applications, IIT Bombay, 2018.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§ 1 Borsuk-Ulam Theorem
A well-known B o r s u k - U l a m t h e o r e m says that:
Theorem 1.1 ( B o r s u k - U l a m )
For every continuous map g : Sn→ Rn, n ∈ N, there existanti-podal points t, −t ∈ Sn with g(t) = g(−t).
Sn = {t = (t0, . . . , tn) ∈ Rn+1 | ||t||2 = ∑ni=0 t2
i = 1} ⊆ Rn+1 is then-sphere.
This was conjectured by S. Ulam and was proved by K. Borsuk in1933 by elementary methods but technically involved.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§ 1 Borsuk-Ulam Theorem
A well-known B o r s u k - U l a m t h e o r e m says that:
Theorem 1.1 ( B o r s u k - U l a m )
For every continuous map g : Sn→ Rn, n ∈ N, there existanti-podal points t, −t ∈ Sn with g(t) = g(−t).
Sn = {t = (t0, . . . , tn) ∈ Rn+1 | ||t||2 = ∑ni=0 t2
i = 1} ⊆ Rn+1 is then-sphere.
This was conjectured by S. Ulam and was proved by K. Borsuk in1933 by elementary methods but technically involved.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§ 1 Borsuk-Ulam Theorem
A well-known B o r s u k - U l a m t h e o r e m says that:
Theorem 1.1 ( B o r s u k - U l a m )
For every continuous map g : Sn→ Rn, n ∈ N, there existanti-podal points t, −t ∈ Sn with g(t) = g(−t).
Sn = {t = (t0, . . . , tn) ∈ Rn+1 | ||t||2 = ∑ni=0 t2
i = 1} ⊆ Rn+1 is then-sphere.
This was conjectured by S. Ulam and was proved by K. Borsuk in1933 by elementary methods but technically involved.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984) Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984)
Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984)
Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984) Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984) Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.
It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Karol Borsuk and Stanislaw Ulam
Stanislaw Ulam (1909 – 1984) Karol Borsuk (1905 – 1982)
Borsuk presented the theorem at the International Congress of Mathenmatics atZürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190(1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre.It was also already remarked by Borsuk in the footnote that H. Hopf also provedthis theorem with the methods of Algebraic topology, mainly using the Theory ofmapping degrees (which goes back to L. E. J. Brouwer). This appeared in thebook of P. Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1.Aufl. : Berlin 1935).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Pavel Aleksandrov and Heinz Hopf
Pavel Aleksandrov (1896–1982) Heinz Hopf (1894–1971)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Pavel Aleksandrov and Heinz Hopf
Pavel Aleksandrov (1896–1982)
Heinz Hopf (1894–1971)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Pavel Aleksandrov and Heinz Hopf
Pavel Aleksandrov (1896–1982)
Heinz Hopf (1894–1971)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Pavel Aleksandrov and Heinz Hopf
Pavel Aleksandrov (1896–1982) Heinz Hopf (1894–1971)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Applications of Borsuk-Ulam Theorem
The Theorem of Borsuk-Ulam is facinating even today. It impliesthe I n v a r i a n c e o f D i m e n s i o n and the classicalT h e o r e m o f B r o u w e r :
Theorem 1.2 ( I n v a r i a n c e o f d i m e n s i o n )
If m 6= n, then the Euclidean spaces Rm and Rn are nothomeomorphic.
Theorem 1.3 ( B r o u w e r ’ s f i x e d p o i n t t h e o r e m )
Every continuous map f : Bn→ Bn of the unit ballBn := {t ∈ Rn | ||t|| ≤ 1} has a fixed point.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Applications of Borsuk-Ulam Theorem
The Theorem of Borsuk-Ulam is facinating even today. It impliesthe I n v a r i a n c e o f D i m e n s i o n and the classicalT h e o r e m o f B r o u w e r :
Theorem 1.2 ( I n v a r i a n c e o f d i m e n s i o n )
If m 6= n, then the Euclidean spaces Rm and Rn are nothomeomorphic.
Theorem 1.3 ( B r o u w e r ’ s f i x e d p o i n t t h e o r e m )
Every continuous map f : Bn→ Bn of the unit ballBn := {t ∈ Rn | ||t|| ≤ 1} has a fixed point.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Applications of Borsuk-Ulam Theorem
The Theorem of Borsuk-Ulam is facinating even today. It impliesthe I n v a r i a n c e o f D i m e n s i o n and the classicalT h e o r e m o f B r o u w e r :
Theorem 1.2 ( I n v a r i a n c e o f d i m e n s i o n )
If m 6= n, then the Euclidean spaces Rm and Rn are nothomeomorphic.
Theorem 1.3 ( B r o u w e r ’ s f i x e d p o i n t t h e o r e m )
Every continuous map f : Bn→ Bn of the unit ballBn := {t ∈ Rn | ||t|| ≤ 1} has a fixed point.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.Schrauder’s fixed point theorem.Ham Sandwich Theorem.
and many more . . .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.Schrauder’s fixed point theorem.Ham Sandwich Theorem.
and many more . . .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.
Schrauder’s fixed point theorem.Ham Sandwich Theorem.
and many more . . .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.Schrauder’s fixed point theorem.
Ham Sandwich Theorem.
and many more . . .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.Schrauder’s fixed point theorem.Ham Sandwich Theorem.
and many more . . .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Brouwer L. E. J. (1881 – 1966)
Brouwer L. E. J. (1881 – 1966)
Many more applications of Borsuk-Ulam Theorem :
Generalization of the Brouwer’s fixed point theorem toBanach spaces.Schrauder’s fixed point theorem.Ham Sandwich Theorem.
and many more . . .Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§2 Borsuk’s Nullstellensatz and its Equivalents
Theorem 2.1
Let n ∈ N. Then the following statements are equivalent:
(i) (B o r s u k - U l a m ’s N u l l s t e l l e n s a t z) Everycontinuous o d d m a p f : Sn→ Rn has a zero.
(ii) (B o r s u k ’ s a n t i p o d a l t h e o r e m) Every continuousmap h : Bn→ Rn with n≥ 1 and h|Sn−1 : Sn−1→ Rn odd,has a zero.
(iii) (R e a l a l g e b r a i c N u l l s t e l l e n s a t z) Homogeneouspolynomials f1, ..., fn ∈ R[T0, ...,Tn] of odd degree have acommon non-trivial zero in Rn+1.
A map f : Sn→ Rn is called an o d d m a p if f (x) =−f (−x) for every x ∈ Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§2 Borsuk’s Nullstellensatz and its Equivalents
Theorem 2.1
Let n ∈ N. Then the following statements are equivalent:
(i) (B o r s u k - U l a m ’s N u l l s t e l l e n s a t z) Everycontinuous o d d m a p f : Sn→ Rn has a zero.
(ii) (B o r s u k ’ s a n t i p o d a l t h e o r e m) Every continuousmap h : Bn→ Rn with n≥ 1 and h|Sn−1 : Sn−1→ Rn odd,has a zero.
(iii) (R e a l a l g e b r a i c N u l l s t e l l e n s a t z) Homogeneouspolynomials f1, ..., fn ∈ R[T0, ...,Tn] of odd degree have acommon non-trivial zero in Rn+1.
A map f : Sn→ Rn is called an o d d m a p if f (x) =−f (−x) for every x ∈ Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§2 Borsuk’s Nullstellensatz and its Equivalents
Theorem 2.1
Let n ∈ N. Then the following statements are equivalent:
(i) (B o r s u k - U l a m ’s N u l l s t e l l e n s a t z) Everycontinuous o d d m a p f : Sn→ Rn has a zero.
(ii) (B o r s u k ’ s a n t i p o d a l t h e o r e m) Every continuousmap h : Bn→ Rn with n≥ 1 and h|Sn−1 : Sn−1→ Rn odd,has a zero.
(iii) (R e a l a l g e b r a i c N u l l s t e l l e n s a t z) Homogeneouspolynomials f1, ..., fn ∈ R[T0, ...,Tn] of odd degree have acommon non-trivial zero in Rn+1.
A map f : Sn→ Rn is called an o d d m a p if f (x) =−f (−x) for every x ∈ Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§2 Borsuk’s Nullstellensatz and its Equivalents
Theorem 2.1
Let n ∈ N. Then the following statements are equivalent:
(i) (B o r s u k - U l a m ’s N u l l s t e l l e n s a t z) Everycontinuous o d d m a p f : Sn→ Rn has a zero.
(ii) (B o r s u k ’ s a n t i p o d a l t h e o r e m) Every continuousmap h : Bn→ Rn with n≥ 1 and h|Sn−1 : Sn−1→ Rn odd,has a zero.
(iii) (R e a l a l g e b r a i c N u l l s t e l l e n s a t z) Homogeneouspolynomials f1, ..., fn ∈ R[T0, ...,Tn] of odd degree have acommon non-trivial zero in Rn+1.
A map f : Sn→ Rn is called an o d d m a p if f (x) =−f (−x) for every x ∈ Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§2 Borsuk’s Nullstellensatz and its Equivalents
Theorem 2.1
Let n ∈ N. Then the following statements are equivalent:
(i) (B o r s u k - U l a m ’s N u l l s t e l l e n s a t z) Everycontinuous o d d m a p f : Sn→ Rn has a zero.
(ii) (B o r s u k ’ s a n t i p o d a l t h e o r e m) Every continuousmap h : Bn→ Rn with n≥ 1 and h|Sn−1 : Sn−1→ Rn odd,has a zero.
(iii) (R e a l a l g e b r a i c N u l l s t e l l e n s a t z) Homogeneouspolynomials f1, ..., fn ∈ R[T0, ...,Tn] of odd degree have acommon non-trivial zero in Rn+1.
A map f : Sn→ Rn is called an o d d m a p if f (x) =−f (−x) for every x ∈ Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Weierstrass Approximation Theorem
Proof of the implication (iii)⇒ (i) use the well-knownWeierstrass Approximation Theorem :
Theorem 2.2 ( We i e r s t r a s s )Let X ⊆ Rn, n ∈ N be a compact subset. Then the set of polynomialfunctions R[T1, ...,Tn] is dense in (C(X,R), || · ||sup), where for everyf ∈ C(X,R), ||f ||sup := Sup{||f (x)|| x ∈ X}.
Karl Weierstrass (1815 – 1897)Karl Weierstrass is known as the father of modern analysis, and contributed tothe theory of periodic functions, functions of real variables, elliptic functions,Abelian functions, converging infinite products, and the calculus of variations.He also advanced the theory of bilinear and quadratic forms.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Weierstrass Approximation Theorem
Proof of the implication (iii)⇒ (i) use the well-knownWeierstrass Approximation Theorem :
Theorem 2.2 ( We i e r s t r a s s )Let X ⊆ Rn, n ∈ N be a compact subset. Then the set of polynomialfunctions R[T1, ...,Tn] is dense in (C(X,R), || · ||sup), where for everyf ∈ C(X,R), ||f ||sup := Sup{||f (x)|| x ∈ X}.
Karl Weierstrass (1815 – 1897)Karl Weierstrass is known as the father of modern analysis, and contributed tothe theory of periodic functions, functions of real variables, elliptic functions,Abelian functions, converging infinite products, and the calculus of variations.He also advanced the theory of bilinear and quadratic forms.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Weierstrass Approximation Theorem
Proof of the implication (iii)⇒ (i) use the well-knownWeierstrass Approximation Theorem :
Theorem 2.2 ( We i e r s t r a s s )Let X ⊆ Rn, n ∈ N be a compact subset. Then the set of polynomialfunctions R[T1, ...,Tn] is dense in (C(X,R), || · ||sup), where for everyf ∈ C(X,R), ||f ||sup := Sup{||f (x)|| x ∈ X}.
Karl Weierstrass (1815 – 1897)
Karl Weierstrass is known as the father of modern analysis, and contributed tothe theory of periodic functions, functions of real variables, elliptic functions,Abelian functions, converging infinite products, and the calculus of variations.He also advanced the theory of bilinear and quadratic forms.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Weierstrass Approximation Theorem
Proof of the implication (iii)⇒ (i) use the well-knownWeierstrass Approximation Theorem :
Theorem 2.2 ( We i e r s t r a s s )Let X ⊆ Rn, n ∈ N be a compact subset. Then the set of polynomialfunctions R[T1, ...,Tn] is dense in (C(X,R), || · ||sup), where for everyf ∈ C(X,R), ||f ||sup := Sup{||f (x)|| x ∈ X}.
Karl Weierstrass (1815 – 1897)Karl Weierstrass is known as the father of modern analysis, and contributed tothe theory of periodic functions, functions of real variables, elliptic functions,Abelian functions, converging infinite products, and the calculus of variations.He also advanced the theory of bilinear and quadratic forms.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Proof of Real Algebraic Nullstellensatz⇒ Borsuk-Ulam’s
Nullstellensatz
Proof of (iii)⇒(i) :
Let f = (f1, . . . , fn) : Sn→ Rn with fi : Sn→ R odd and continuous. Bythe Weierstrass Approximation Theorem for every k ∈ N∗, there existpolynomial functions gik with |gik(t)− fi(t)| ≤ 1/k for i = 1, . . . ,n and allt ∈ Sn. For the odd parts fik(t) := (gik(t)−gik(−t))/2, it follows|fik(t)− fi(t)|= 1
2 |(gik(t)− fi(t))− (gik(−t)− fi(−t))| ≤ 1/k. By the realalgebraic Nullstellensatz, the fik, i = 1, . . . ,n, have a common zerotk ∈ Sn. Then an accumulation point t ∈ Sn of tk, k ∈ N∗, is a commonzero of the f1, . . . , fn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Proof of Real Algebraic Nullstellensatz⇒ Borsuk-Ulam’s
Nullstellensatz
Proof of (iii)⇒(i) :
Let f = (f1, . . . , fn) : Sn→ Rn with fi : Sn→ R odd and continuous. Bythe Weierstrass Approximation Theorem for every k ∈ N∗, there existpolynomial functions gik with |gik(t)− fi(t)| ≤ 1/k for i = 1, . . . ,n and allt ∈ Sn. For the odd parts fik(t) := (gik(t)−gik(−t))/2, it follows|fik(t)− fi(t)|= 1
2 |(gik(t)− fi(t))− (gik(−t)− fi(−t))| ≤ 1/k. By the realalgebraic Nullstellensatz, the fik, i = 1, . . . ,n, have a common zerotk ∈ Sn. Then an accumulation point t ∈ Sn of tk, k ∈ N∗, is a commonzero of the f1, . . . , fn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk-Ulam’s Nullstellensatz⇒ Real Algebraic Nullstellensatz
For a proof of the implication (i)⇒ (iii) use the following trick :
From (i) in particular, it follows that n odd polynomial functionsfi : Rn+1→ R have a common zero on Sn.
If F = F1 +F3 + · · ·+F2m+1, F2m+1 6= 0, is the decomposition ofan odd polynomial F ∈ R[T0, . . . ,Tn] of degree 2m+1 in itshomogeneous components, then F and the homogeneouspolynomial QmF1 +Qm−1F3 + · · ·+F2m+1, Q := T2
0 + · · ·+T2n ,
have the same values on the sphere Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk-Ulam’s Nullstellensatz⇒ Real Algebraic Nullstellensatz
For a proof of the implication (i)⇒ (iii) use the following trick :
From (i) in particular, it follows that n odd polynomial functionsfi : Rn+1→ R have a common zero on Sn.
If F = F1 +F3 + · · ·+F2m+1, F2m+1 6= 0, is the decomposition ofan odd polynomial F ∈ R[T0, . . . ,Tn] of degree 2m+1 in itshomogeneous components, then F and the homogeneouspolynomial QmF1 +Qm−1F3 + · · ·+F2m+1, Q := T2
0 + · · ·+T2n ,
have the same values on the sphere Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk-Ulam’s Nullstellensatz⇒ Real Algebraic Nullstellensatz
For a proof of the implication (i)⇒ (iii) use the following trick :
From (i) in particular, it follows that n odd polynomial functionsfi : Rn+1→ R have a common zero on Sn.
If F = F1 +F3 + · · ·+F2m+1, F2m+1 6= 0, is the decomposition ofan odd polynomial F ∈ R[T0, . . . ,Tn] of degree 2m+1 in itshomogeneous components, then F and the homogeneouspolynomial QmF1 +Qm−1F3 + · · ·+F2m+1, Q := T2
0 + · · ·+T2n ,
have the same values on the sphere Sn.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk’s Nullstellensatz⇒ Borsuk-Ulam Theorem
Corollary 2.3 ( B o r s u k - U l a m T h e o r e m )
For every continuous map g : Sn→ Rn , there exists t ∈ Sn suchthat g(t) = g(−t).
Proof Consider the off continuous map f : Sn→ Rn,t 7→ f (t) := g(t)−g(−t) and use (i) in Theorem 2.1.
The Real algebraic Nullstellensatz in Theorem 2.1-iii reminds usthe famous Hilbert’s Nullstellensatz :
Theorem 2.4 ( H i l b e r t ’ s N u l s t e l l e n s a t z )
Homogeneous polynomial polynomials f1, ..., fn ∈ K[T0, ...,Tn] ofarbitrary positive degrees over an algebraically closed field K,have a common non-trivial zero in Kn+1. (for example, overK = C.)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk’s Nullstellensatz⇒ Borsuk-Ulam Theorem
Corollary 2.3 ( B o r s u k - U l a m T h e o r e m )
For every continuous map g : Sn→ Rn , there exists t ∈ Sn suchthat g(t) = g(−t).
Proof Consider the off continuous map f : Sn→ Rn,t 7→ f (t) := g(t)−g(−t) and use (i) in Theorem 2.1.
The Real algebraic Nullstellensatz in Theorem 2.1-iii reminds usthe famous Hilbert’s Nullstellensatz :
Theorem 2.4 ( H i l b e r t ’ s N u l s t e l l e n s a t z )
Homogeneous polynomial polynomials f1, ..., fn ∈ K[T0, ...,Tn] ofarbitrary positive degrees over an algebraically closed field K,have a common non-trivial zero in Kn+1. (for example, overK = C.)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk’s Nullstellensatz⇒ Borsuk-Ulam Theorem
Corollary 2.3 ( B o r s u k - U l a m T h e o r e m )
For every continuous map g : Sn→ Rn , there exists t ∈ Sn suchthat g(t) = g(−t).
Proof Consider the off continuous map f : Sn→ Rn,t 7→ f (t) := g(t)−g(−t) and use (i) in Theorem 2.1.
The Real algebraic Nullstellensatz in Theorem 2.1-iii reminds usthe famous Hilbert’s Nullstellensatz :
Theorem 2.4 ( H i l b e r t ’ s N u l s t e l l e n s a t z )
Homogeneous polynomial polynomials f1, ..., fn ∈ K[T0, ...,Tn] ofarbitrary positive degrees over an algebraically closed field K,have a common non-trivial zero in Kn+1. (for example, overK = C.)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Borsuk’s Nullstellensatz⇒ Borsuk-Ulam Theorem
Corollary 2.3 ( B o r s u k - U l a m T h e o r e m )
For every continuous map g : Sn→ Rn , there exists t ∈ Sn suchthat g(t) = g(−t).
Proof Consider the off continuous map f : Sn→ Rn,t 7→ f (t) := g(t)−g(−t) and use (i) in Theorem 2.1.
The Real algebraic Nullstellensatz in Theorem 2.1-iii reminds usthe famous Hilbert’s Nullstellensatz :
Theorem 2.4 ( H i l b e r t ’ s N u l s t e l l e n s a t z )
Homogeneous polynomial polynomials f1, ..., fn ∈ K[T0, ...,Tn] ofarbitrary positive degrees over an algebraically closed field K,have a common non-trivial zero in Kn+1. (for example, overK = C.)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
David Hilbert
David Hilbert (1862 – 1943)
Hilbert proved Nullstellensatz (for K = C) in : Über die vollenInvariantenstyteme, Math. Ann. 42, 313-373 (1893).
Hilbert’s famous 23 Paris problems challenged (and still today challenge)mathematicians to solve fundamental questions. Hilbert’s famous speech TheProblems of Mathematics was delivered at the Second International Congressof Mathematicians in Paris in 1900. It was a speech full of optimism formathematics in the coming century and he felt that open problems were the signof vitality in the subject.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
David Hilbert
David Hilbert (1862 – 1943)
Hilbert proved Nullstellensatz (for K = C) in : Über die vollenInvariantenstyteme, Math. Ann. 42, 313-373 (1893).
Hilbert’s famous 23 Paris problems challenged (and still today challenge)mathematicians to solve fundamental questions. Hilbert’s famous speech TheProblems of Mathematics was delivered at the Second International Congressof Mathematicians in Paris in 1900. It was a speech full of optimism formathematics in the coming century and he felt that open problems were the signof vitality in the subject.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
David Hilbert
David Hilbert (1862 – 1943)
Hilbert proved Nullstellensatz (for K = C) in : Über die vollenInvariantenstyteme, Math. Ann. 42, 313-373 (1893).
Hilbert’s famous 23 Paris problems challenged (and still today challenge)mathematicians to solve fundamental questions. Hilbert’s famous speech TheProblems of Mathematics was delivered at the Second International Congressof Mathematicians in Paris in 1900. It was a speech full of optimism formathematics in the coming century and he felt that open problems were the signof vitality in the subject.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Intermediate Value theorem
The only property of the field R of real numbers used is :
Theorem 2.5
Every polynomial F ∈ R[X] of odd degree has a zero in R.
This is an immediate consequence of the Intermediate Value Theoremfor continuous functions R→ R :
Theorem 2.6 ( I n t e r m e d i a t e Va l u e T h e o r e m )
Let f : R→ R be a real-valued continuous function and a , b ∈ R ,a < b. For every c in between f (a) and f (b), there exists t0 ∈ [a , b ] withf (t0) = c.The intermediate value theorem characterizes the completeness property in anordered field : Suppose that every continuous function f : K→ K takes positiveas well as negative values, has a zero in K. Then K is a field of real numbers.Therefore K ∼= R.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Intermediate Value theorem
The only property of the field R of real numbers used is :
Theorem 2.5
Every polynomial F ∈ R[X] of odd degree has a zero in R.
This is an immediate consequence of the Intermediate Value Theoremfor continuous functions R→ R :
Theorem 2.6 ( I n t e r m e d i a t e Va l u e T h e o r e m )
Let f : R→ R be a real-valued continuous function and a , b ∈ R ,a < b. For every c in between f (a) and f (b), there exists t0 ∈ [a , b ] withf (t0) = c.The intermediate value theorem characterizes the completeness property in anordered field : Suppose that every continuous function f : K→ K takes positiveas well as negative values, has a zero in K. Then K is a field of real numbers.Therefore K ∼= R.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Intermediate Value theorem
The only property of the field R of real numbers used is :
Theorem 2.5
Every polynomial F ∈ R[X] of odd degree has a zero in R.
This is an immediate consequence of the Intermediate Value Theoremfor continuous functions R→ R :
Theorem 2.6 ( I n t e r m e d i a t e Va l u e T h e o r e m )
Let f : R→ R be a real-valued continuous function and a , b ∈ R ,a < b. For every c in between f (a) and f (b), there exists t0 ∈ [a , b ] withf (t0) = c.The intermediate value theorem characterizes the completeness property in anordered field : Suppose that every continuous function f : K→ K takes positiveas well as negative values, has a zero in K. Then K is a field of real numbers.Therefore K ∼= R.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Intermediate Value theorem
The only property of the field R of real numbers used is :
Theorem 2.5
Every polynomial F ∈ R[X] of odd degree has a zero in R.
This is an immediate consequence of the Intermediate Value Theoremfor continuous functions R→ R :
Theorem 2.6 ( I n t e r m e d i a t e Va l u e T h e o r e m )
Let f : R→ R be a real-valued continuous function and a , b ∈ R ,a < b. For every c in between f (a) and f (b), there exists t0 ∈ [a , b ] withf (t0) = c.
The intermediate value theorem characterizes the completeness property in anordered field : Suppose that every continuous function f : K→ K takes positiveas well as negative values, has a zero in K. Then K is a field of real numbers.Therefore K ∼= R.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Intermediate Value theorem
The only property of the field R of real numbers used is :
Theorem 2.5
Every polynomial F ∈ R[X] of odd degree has a zero in R.
This is an immediate consequence of the Intermediate Value Theoremfor continuous functions R→ R :
Theorem 2.6 ( I n t e r m e d i a t e Va l u e T h e o r e m )
Let f : R→ R be a real-valued continuous function and a , b ∈ R ,a < b. For every c in between f (a) and f (b), there exists t0 ∈ [a , b ] withf (t0) = c.The intermediate value theorem characterizes the completeness property in anordered field : Suppose that every continuous function f : K→ K takes positiveas well as negative values, has a zero in K. Then K is a field of real numbers.Therefore K ∼= R.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848) Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e. The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848) Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e. The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848)
Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e. The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848)
Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e.
The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848)
Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e.
The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848) Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e.
The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848) Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e. The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.
Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Bernard Bolzano
The validity of Theorem 2.6 belongs to the foundations of real analysis and hasbeen considered at all times as an obvious fact though a first convincing proof ofit (for continuous functions) was given by B. Bolzano only in 1817.
Bernard Bolzano (1781 – 1848) Augustin Louis Cauchy (1789 – 1857)
Bolzano proved the intermediate value theorem in 1817 and he defined what isnow called a C a u c h y s e q u e n c e. The concept appears in Cauchy’s workfour years later but it is unlikely (?) that Cauchy had read Bolzano’s work.Bolzano gave examples of 1-1 correspondences between the elements of aninfinite set and the elements of a proper subset.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields (recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields (recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields (recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields
(recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields (recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§3 2-Fields
We axiomatize the property in Theorem 2.5 algebraically explicitand define :
Definition 3.1
A field K is called a 2 - f i e l d if every polynomial F ∈ K[X] ofodd degree has a zero in K.
For example, the fields R and C of real and complex numbers are2-fields. More generally, algebraically closed fields are 2-fields.
All real-closed fields (recall that a field K is real-closed if and only ifthe algebraic closure K of K has degree 2 over K.) are 2-fields, see???.
For more examples, we use the following elementarycharacterization of 2-fields :
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Characterization of 2-fields
Lemma 3.2
For a field K, the following statements are equivalent :
(i) K is a 2-field.
(ii) If π ∈ K[X] is a prime polynomial of degree > 1, then thedegree degπ is even.
(iii) If L |K is a non-trivial finite field extension of K, then thedegree = [L : K] = Dim K L is even.
Definition 3.3
A finite field extension L |K is called a 2 - e x t e n s i o n if itsdegree [L : K] is a power of 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Characterization of 2-fields
Lemma 3.2
For a field K, the following statements are equivalent :
(i) K is a 2-field.
(ii) If π ∈ K[X] is a prime polynomial of degree > 1, then thedegree degπ is even.
(iii) If L |K is a non-trivial finite field extension of K, then thedegree = [L : K] = Dim K L is even.
Definition 3.3
A finite field extension L |K is called a 2 - e x t e n s i o n if itsdegree [L : K] is a power of 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Characterization of 2-fields
Lemma 3.2
For a field K, the following statements are equivalent :
(i) K is a 2-field.
(ii) If π ∈ K[X] is a prime polynomial of degree > 1, then thedegree degπ is even.
(iii) If L |K is a non-trivial finite field extension of K, then thedegree = [L : K] = Dim K L is even.
Definition 3.3
A finite field extension L |K is called a 2 - e x t e n s i o n if itsdegree [L : K] is a power of 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Characterization of 2-fields
Lemma 3.2
For a field K, the following statements are equivalent :
(i) K is a 2-field.
(ii) If π ∈ K[X] is a prime polynomial of degree > 1, then thedegree degπ is even.
(iii) If L |K is a non-trivial finite field extension of K, then thedegree = [L : K] = Dim K L is even.
Definition 3.3
A finite field extension L |K is called a 2 - e x t e n s i o n if itsdegree [L : K] is a power of 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Characterization of 2-fields
Lemma 3.2
For a field K, the following statements are equivalent :
(i) K is a 2-field.
(ii) If π ∈ K[X] is a prime polynomial of degree > 1, then thedegree degπ is even.
(iii) If L |K is a non-trivial finite field extension of K, then thedegree = [L : K] = Dim K L is even.
Definition 3.3
A finite field extension L |K is called a 2 - e x t e n s i o n if itsdegree [L : K] is a power of 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
2-fields contd ...
One can use Fundamental Theorem of Galois Theory to prove :
Theorem 3.4
Let K be a 2-field. Then every finite field extension L |K is a2-extension. In particular, every algebraic field extension of K is a2-field.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
2-fields contd ...
One can use Fundamental Theorem of Galois Theory to prove :
Theorem 3.4
Let K be a 2-field. Then every finite field extension L |K is a2-extension. In particular, every algebraic field extension of K is a2-field.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
2-fields contd ...
One can use Fundamental Theorem of Galois Theory to prove :
Theorem 3.4
Let K be a 2-field. Then every finite field extension L |K is a2-extension. In particular, every algebraic field extension of K is a2-field.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832) Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832)
Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832)
Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.
Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832)
Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.
Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832) Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.
Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Galois Theory — Fundamental Theorem of Galois Theory
Évariste Galois and Niels Henrik Abel
Évariste Galois (1811 – 1832) Niels Henrik Abel (1802 – 1829)
Évariste Galois produced a method of determining when a general equationcould be solved by radicals and is famous for his development of early grouptheory.Niels Abel was a Norwegian mathematician who proved the impossibility ofsolving algebraically the general equation of the fifth degree.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Constructible Field extensions
2-extensions play an important roll in the constructions bystraight-edge and compass.
Constructible field extensions are separable and its Galois-hull is afinite 2-extension. Moreover, the converse also hold by Galoistheory (together with group theory). Therefore :
Theorem 3.5
A field K is a 2-field if and only if for every finite field extensionL |K, there exists a chain of fields K = L0 ( L1 ( · · ·( Ls = Lwith [Li : Li−1] = 2, i = 1, . . . ,s.
Corollary 3.6
A 2-field K is algebraically closed if and only if K has noquadratic field extension L |K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Constructible Field extensions
2-extensions play an important roll in the constructions bystraight-edge and compass.
Constructible field extensions are separable and its Galois-hull is afinite 2-extension. Moreover, the converse also hold by Galoistheory (together with group theory). Therefore :
Theorem 3.5
A field K is a 2-field if and only if for every finite field extensionL |K, there exists a chain of fields K = L0 ( L1 ( · · ·( Ls = Lwith [Li : Li−1] = 2, i = 1, . . . ,s.
Corollary 3.6
A 2-field K is algebraically closed if and only if K has noquadratic field extension L |K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Constructible Field extensions
2-extensions play an important roll in the constructions bystraight-edge and compass.
Constructible field extensions are separable and its Galois-hull is afinite 2-extension. Moreover, the converse also hold by Galoistheory (together with group theory). Therefore :
Theorem 3.5
A field K is a 2-field if and only if for every finite field extensionL |K, there exists a chain of fields K = L0 ( L1 ( · · ·( Ls = Lwith [Li : Li−1] = 2, i = 1, . . . ,s.
Corollary 3.6
A 2-field K is algebraically closed if and only if K has noquadratic field extension L |K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Constructible Field extensions
2-extensions play an important roll in the constructions bystraight-edge and compass.
Constructible field extensions are separable and its Galois-hull is afinite 2-extension. Moreover, the converse also hold by Galoistheory (together with group theory). Therefore :
Theorem 3.5
A field K is a 2-field if and only if for every finite field extensionL |K, there exists a chain of fields K = L0 ( L1 ( · · ·( Ls = Lwith [Li : Li−1] = 2, i = 1, . . . ,s.
Corollary 3.6
A 2-field K is algebraically closed if and only if K has noquadratic field extension L |K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Fundamental Theorem of Algebra
From the Corollary 3.6, the known elegant proof due to E. Artinfor the following assertion is immediate :
Theorem 3.7 ( A r t i n - S c h r e i e r )
A field K is a 2-field with the following property : −1 is not asquare in K and in the quadratic extension K[
√−1], every
element is a square. Then K[√−1] is algebraically closed.
Theorem 3.7 implies :
Theorem 3.8 ( F u n d a m e n t a l t h e o r e m o f A l g e b r a )
The field C of Complex numbers is algebraically closed.
The Fundamental Theorem of Algebra 3.8 was stated first time in 1746 byD’Alembert. C. F. Gauss proved in his doctoral thesis from 1799 alsoThereom 3.11 without using complex numbers, but in a completely differentway. All these proofs have (serious) gaps.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Fundamental Theorem of Algebra
From the Corollary 3.6, the known elegant proof due to E. Artinfor the following assertion is immediate :
Theorem 3.7 ( A r t i n - S c h r e i e r )
A field K is a 2-field with the following property : −1 is not asquare in K and in the quadratic extension K[
√−1], every
element is a square. Then K[√−1] is algebraically closed.
Theorem 3.7 implies :
Theorem 3.8 ( F u n d a m e n t a l t h e o r e m o f A l g e b r a )
The field C of Complex numbers is algebraically closed.
The Fundamental Theorem of Algebra 3.8 was stated first time in 1746 byD’Alembert. C. F. Gauss proved in his doctoral thesis from 1799 alsoThereom 3.11 without using complex numbers, but in a completely differentway. All these proofs have (serious) gaps.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Fundamental Theorem of Algebra
From the Corollary 3.6, the known elegant proof due to E. Artinfor the following assertion is immediate :
Theorem 3.7 ( A r t i n - S c h r e i e r )
A field K is a 2-field with the following property : −1 is not asquare in K and in the quadratic extension K[
√−1], every
element is a square. Then K[√−1] is algebraically closed.
Theorem 3.7 implies :
Theorem 3.8 ( F u n d a m e n t a l t h e o r e m o f A l g e b r a )
The field C of Complex numbers is algebraically closed.
The Fundamental Theorem of Algebra 3.8 was stated first time in 1746 byD’Alembert. C. F. Gauss proved in his doctoral thesis from 1799 alsoThereom 3.11 without using complex numbers, but in a completely differentway. All these proofs have (serious) gaps.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Fundamental Theorem of Algebra
From the Corollary 3.6, the known elegant proof due to E. Artinfor the following assertion is immediate :
Theorem 3.7 ( A r t i n - S c h r e i e r )
A field K is a 2-field with the following property : −1 is not asquare in K and in the quadratic extension K[
√−1], every
element is a square. Then K[√−1] is algebraically closed.
Theorem 3.7 implies :
Theorem 3.8 ( F u n d a m e n t a l t h e o r e m o f A l g e b r a )
The field C of Complex numbers is algebraically closed.
The Fundamental Theorem of Algebra 3.8 was stated first time in 1746 byD’Alembert. C. F. Gauss proved in his doctoral thesis from 1799 alsoThereom 3.11 without using complex numbers, but in a completely differentway. All these proofs have (serious) gaps.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Fundamental Theorem of Algebra
From the Corollary 3.6, the known elegant proof due to E. Artinfor the following assertion is immediate :
Theorem 3.7 ( A r t i n - S c h r e i e r )
A field K is a 2-field with the following property : −1 is not asquare in K and in the quadratic extension K[
√−1], every
element is a square. Then K[√−1] is algebraically closed.
Theorem 3.7 implies :
Theorem 3.8 ( F u n d a m e n t a l t h e o r e m o f A l g e b r a )
The field C of Complex numbers is algebraically closed.
The Fundamental Theorem of Algebra 3.8 was stated first time in 1746 byD’Alembert. C. F. Gauss proved in his doctoral thesis from 1799 alsoThereom 3.11 without using complex numbers, but in a completely differentway. All these proofs have (serious) gaps.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emil Artin and Otto Schreier
Emil Artin (1898 – 1962) Otto Schreier (1901 – 1929)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emil Artin and Otto Schreier
Emil Artin (1898 – 1962)
Otto Schreier (1901 – 1929)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emil Artin and Otto Schreier
Emil Artin (1898 – 1962)
Otto Schreier (1901 – 1929)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emil Artin and Otto Schreier
Emil Artin (1898 – 1962) Otto Schreier (1901 – 1929)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emil Artin and Otto Schreier
Emil Artin (1898 – 1962) Otto Schreier (1901 – 1929)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
History of FTA
The IVT (Theorem 2.6) directly implies the I V T f o r r e a lp o l y n o m i a l s:
Theorem 3.9 ( I V T — f o r r e a l p o l y n o m i a l s )If the values F(a) and F(b) of the real polynomial function F havedifferent signs at a , b ∈ R , a < b, then F has a zero in the interval[a , b ] .
Theorem 3.9 may be considered as the equivalent to the following :
Theorem 3.10 ( F TA — C o m p l e x v e r s i o n )Every non-constant complex polynomial function C→ C is surjective.
Theorem 3.10 is equivalent to the FTA, but, it is far from being obviousthat Theorem 3.9 implies the (real) FTA :
Theorem 3.11 ( F TA — R e a l v e r s i o n )Every non-constant real polynomial is divisible by a (monic) realpolynomial of degree 1 or 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
History of FTA
The IVT (Theorem 2.6) directly implies the I V T f o r r e a lp o l y n o m i a l s:
Theorem 3.9 ( I V T — f o r r e a l p o l y n o m i a l s )If the values F(a) and F(b) of the real polynomial function F havedifferent signs at a , b ∈ R , a < b, then F has a zero in the interval[a , b ] .
Theorem 3.9 may be considered as the equivalent to the following :
Theorem 3.10 ( F TA — C o m p l e x v e r s i o n )Every non-constant complex polynomial function C→ C is surjective.
Theorem 3.10 is equivalent to the FTA, but, it is far from being obviousthat Theorem 3.9 implies the (real) FTA :
Theorem 3.11 ( F TA — R e a l v e r s i o n )Every non-constant real polynomial is divisible by a (monic) realpolynomial of degree 1 or 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
History of FTA
The IVT (Theorem 2.6) directly implies the I V T f o r r e a lp o l y n o m i a l s:
Theorem 3.9 ( I V T — f o r r e a l p o l y n o m i a l s )If the values F(a) and F(b) of the real polynomial function F havedifferent signs at a , b ∈ R , a < b, then F has a zero in the interval[a , b ] .
Theorem 3.9 may be considered as the equivalent to the following :
Theorem 3.10 ( F TA — C o m p l e x v e r s i o n )Every non-constant complex polynomial function C→ C is surjective.
Theorem 3.10 is equivalent to the FTA, but, it is far from being obviousthat Theorem 3.9 implies the (real) FTA :
Theorem 3.11 ( F TA — R e a l v e r s i o n )Every non-constant real polynomial is divisible by a (monic) realpolynomial of degree 1 or 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
History of FTA
The IVT (Theorem 2.6) directly implies the I V T f o r r e a lp o l y n o m i a l s:
Theorem 3.9 ( I V T — f o r r e a l p o l y n o m i a l s )If the values F(a) and F(b) of the real polynomial function F havedifferent signs at a , b ∈ R , a < b, then F has a zero in the interval[a , b ] .
Theorem 3.9 may be considered as the equivalent to the following :
Theorem 3.10 ( F TA — C o m p l e x v e r s i o n )Every non-constant complex polynomial function C→ C is surjective.
Theorem 3.10 is equivalent to the FTA, but, it is far from being obviousthat Theorem 3.9 implies the (real) FTA :
Theorem 3.11 ( F TA — R e a l v e r s i o n )Every non-constant real polynomial is divisible by a (monic) realpolynomial of degree 1 or 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
History of FTA
The IVT (Theorem 2.6) directly implies the I V T f o r r e a lp o l y n o m i a l s:
Theorem 3.9 ( I V T — f o r r e a l p o l y n o m i a l s )If the values F(a) and F(b) of the real polynomial function F havedifferent signs at a , b ∈ R , a < b, then F has a zero in the interval[a , b ] .
Theorem 3.9 may be considered as the equivalent to the following :
Theorem 3.10 ( F TA — C o m p l e x v e r s i o n )Every non-constant complex polynomial function C→ C is surjective.
Theorem 3.10 is equivalent to the FTA, but, it is far from being obviousthat Theorem 3.9 implies the (real) FTA :
Theorem 3.11 ( F TA — R e a l v e r s i o n )Every non-constant real polynomial is divisible by a (monic) realpolynomial of degree 1 or 2.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783) Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783)
Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783)
Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783) Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783) Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.
Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Jean d’Alembert and Carl Friedrich Gauss
Jean d’Alembert (1717 – 1783) Carl Friedrich Gauss (1777 – 1855)
Jean d’Alembert was a pioneer in the study of differential equations and theiruse of in physics. He studied the equilibrium and motion of fluids.Gauss worked in a wide variety of fields in both mathematics and physicsincluding number theory, analysis, differential geometry, geodesy, magnetism,astronomy and optics. His work has had an immense influence in many areas.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Leonhard Euler
Leonhard Euler (1707 – 1783)
In response to d’Alembert’s proof of the FTA, published in 1748, Euler wrote anarticle (see : [J.-B. le Rond d’Alembert: Recherches sur le calcul intégral. In:Histoire de l’Académie Royale des Sciences et Belle Lettres, AnnéeMDCCXLVI, Berlin, 1748, 182-224.]) which appeared in 1751 and contains(besides many other things) another proof of (real) FTA 3.11. Euler acceptedd’Alembert’s proof completely, but he wanted to provide a more algebraic proof.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Leonhard Euler
Leonhard Euler (1707 – 1783)In response to d’Alembert’s proof of the FTA, published in 1748, Euler wrote anarticle
(see : [J.-B. le Rond d’Alembert: Recherches sur le calcul intégral. In:Histoire de l’Académie Royale des Sciences et Belle Lettres, AnnéeMDCCXLVI, Berlin, 1748, 182-224.]) which appeared in 1751 and contains(besides many other things) another proof of (real) FTA 3.11. Euler acceptedd’Alembert’s proof completely, but he wanted to provide a more algebraic proof.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Leonhard Euler
Leonhard Euler (1707 – 1783)In response to d’Alembert’s proof of the FTA, published in 1748, Euler wrote anarticle (see : [J.-B. le Rond d’Alembert: Recherches sur le calcul intégral. In:Histoire de l’Académie Royale des Sciences et Belle Lettres, AnnéeMDCCXLVI, Berlin, 1748, 182-224.])
which appeared in 1751 and contains(besides many other things) another proof of (real) FTA 3.11. Euler acceptedd’Alembert’s proof completely, but he wanted to provide a more algebraic proof.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Leonhard Euler
Leonhard Euler (1707 – 1783)In response to d’Alembert’s proof of the FTA, published in 1748, Euler wrote anarticle (see : [J.-B. le Rond d’Alembert: Recherches sur le calcul intégral. In:Histoire de l’Académie Royale des Sciences et Belle Lettres, AnnéeMDCCXLVI, Berlin, 1748, 182-224.]) which appeared in 1751 and contains(besides many other things) another proof of (real) FTA 3.11. Euler acceptedd’Alembert’s proof completely, but he wanted to provide a more algebraic proof.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
p-Fields
Analogous to the concept of the 2-field, one can also define the conceptof the p - f i e l d for every prime number p as follows :
Definition 3.12A field K with the property : Every polynomial in F ∈ K[X] of degF notdivisible by p has a zero in K.
Remark 3.13With this definition Lemma 3.2 and Theorem 3.4 also hold if we replace“even” by “divisible by p” (and “odd” by “not divisible by p”).Moreover, one can also study “p-constructibility”.
However, there is no analog for p 6= 2 for the following Theorem :
Theorem 3.14If the algebraic closure K of a field K is finite over K, then K = K or[K : K] = 2, i. e. K is algebraically closed or real-closed.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
p-Fields
Analogous to the concept of the 2-field, one can also define the conceptof the p - f i e l d for every prime number p as follows :
Definition 3.12A field K with the property : Every polynomial in F ∈ K[X] of degF notdivisible by p has a zero in K.
Remark 3.13With this definition Lemma 3.2 and Theorem 3.4 also hold if we replace“even” by “divisible by p” (and “odd” by “not divisible by p”).Moreover, one can also study “p-constructibility”.
However, there is no analog for p 6= 2 for the following Theorem :
Theorem 3.14If the algebraic closure K of a field K is finite over K, then K = K or[K : K] = 2, i. e. K is algebraically closed or real-closed.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
p-Fields
Analogous to the concept of the 2-field, one can also define the conceptof the p - f i e l d for every prime number p as follows :
Definition 3.12A field K with the property : Every polynomial in F ∈ K[X] of degF notdivisible by p has a zero in K.
Remark 3.13With this definition Lemma 3.2 and Theorem 3.4 also hold if we replace“even” by “divisible by p” (and “odd” by “not divisible by p”).Moreover, one can also study “p-constructibility”.
However, there is no analog for p 6= 2 for the following Theorem :
Theorem 3.14If the algebraic closure K of a field K is finite over K, then K = K or[K : K] = 2, i. e. K is algebraically closed or real-closed.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
p-Fields
Analogous to the concept of the 2-field, one can also define the conceptof the p - f i e l d for every prime number p as follows :
Definition 3.12A field K with the property : Every polynomial in F ∈ K[X] of degF notdivisible by p has a zero in K.
Remark 3.13With this definition Lemma 3.2 and Theorem 3.4 also hold if we replace“even” by “divisible by p” (and “odd” by “not divisible by p”).Moreover, one can also study “p-constructibility”.
However, there is no analog for p 6= 2 for the following Theorem :
Theorem 3.14If the algebraic closure K of a field K is finite over K, then K = K or[K : K] = 2, i. e. K is algebraically closed or real-closed.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra. If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra. If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra.
If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra.
If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.
A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra. If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective
and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.
A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra. If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε .
In particular, Am = 0 for m < 0.
A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§4 Dimension and MultiplicityGraded Rings
Let K be a field and let P := K[T0, . . . ,Tn] be the polynomialalgebra. The homogeneous polynomials of degree m ∈ N form aK-subspace Pm of P and P =
⊕m∈N Pm. Further, PkP` ⊆ Pk+` for
all k, ` ∈ N.
More generally, let A =⊕
m∈Z Am be a Z-graded ring. We shallassume that A is a standard graded K-algebra. If t0, . . . , tn ∈ A1 isa generating system of A = K[t0, . . . , tn], then the K-algebrasubstitution homomorphism ε : K[T0, . . . ,Tn]→ A with Ti→ ti,i = 0, . . . ,n, is homogeneous and surjective and so A isisomorphic to the residue algebra K[T0, . . . ,Tn]/A ofP = K[T0, . . . ,Tn] modulo the homogeneous relation idealA := Kerε . In particular, Am = 0 for m < 0.A Z-graded ring A =⊕m∈ZAm is called s t a n d a r d g r a d e d K - a l g e b r a ifA0 = K and A is generated by finitely many homogeneous elements of degree 1as K-algebra.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Graded modules
Let M =⊕
m∈Z Mm be a graded A-module. Every A-module isalso P-module via ε : K[T0, . . . ,Tn]→ A with Ti→ ti, i = 0, . . . ,n,
We shall assume that M is a finite A-module, i. e. finitelygenerated. If x1, . . . ,xr is a homogeneous system of generators ofM of degrees δ1, . . . ,δr ∈ Z, then the canonical homomorphism
A(−δ1)⊕·· ·⊕A(−δr)→M , ei 7→ xi , i = 1, . . . ,r ,
is homogeneous of degree 0 and surjective. The standard basiselement ei ∈ A(−δi) has degree δi.
Theorem 4.1
If A is a standard graded K-algebra and if the graded A-moduleM is finite, then M is a noetherian A-module.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Graded modules
Let M =⊕
m∈Z Mm be a graded A-module. Every A-module isalso P-module via ε : K[T0, . . . ,Tn]→ A with Ti→ ti, i = 0, . . . ,n,
We shall assume that M is a finite A-module, i. e. finitelygenerated.
If x1, . . . ,xr is a homogeneous system of generators ofM of degrees δ1, . . . ,δr ∈ Z, then the canonical homomorphism
A(−δ1)⊕·· ·⊕A(−δr)→M , ei 7→ xi , i = 1, . . . ,r ,
is homogeneous of degree 0 and surjective. The standard basiselement ei ∈ A(−δi) has degree δi.
Theorem 4.1
If A is a standard graded K-algebra and if the graded A-moduleM is finite, then M is a noetherian A-module.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Graded modules
Let M =⊕
m∈Z Mm be a graded A-module. Every A-module isalso P-module via ε : K[T0, . . . ,Tn]→ A with Ti→ ti, i = 0, . . . ,n,
We shall assume that M is a finite A-module, i. e. finitelygenerated. If x1, . . . ,xr is a homogeneous system of generators ofM of degrees δ1, . . . ,δr ∈ Z, then the canonical homomorphism
A(−δ1)⊕·· ·⊕A(−δr)→M , ei 7→ xi , i = 1, . . . ,r ,
is homogeneous of degree 0 and surjective. The standard basiselement ei ∈ A(−δi) has degree δi.
Theorem 4.1
If A is a standard graded K-algebra and if the graded A-moduleM is finite, then M is a noetherian A-module.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Graded modules
Let M =⊕
m∈Z Mm be a graded A-module. Every A-module isalso P-module via ε : K[T0, . . . ,Tn]→ A with Ti→ ti, i = 0, . . . ,n,
We shall assume that M is a finite A-module, i. e. finitelygenerated. If x1, . . . ,xr is a homogeneous system of generators ofM of degrees δ1, . . . ,δr ∈ Z, then the canonical homomorphism
A(−δ1)⊕·· ·⊕A(−δr)→M , ei 7→ xi , i = 1, . . . ,r ,
is homogeneous of degree 0 and surjective. The standard basiselement ei ∈ A(−δi) has degree δi.
Theorem 4.1
If A is a standard graded K-algebra and if the graded A-moduleM is finite, then M is a noetherian A-module.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Noetherian Modules
Emmy Noether
An A-module is called n o e t h e r i a n if every A-submodule of M isfinite. This is equivalent to the condition that M has no infinite properascending chain M0 ( M1 ( M2 ( · · · ⊆M of A-submodules, orequivalently, every non-empty set of A-submodules of M has a maximalelement (with respect to the inclusion).
Emmy Noether (1882 – 1935)
The noetherian property of modules is named after Emmy Noether who is bestknown for her contributions to abstract algebra, in particular, her study of chainconditions on ideals of rings. She was a daughter of Max Noether.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Noetherian Modules
Emmy Noether
An A-module is called n o e t h e r i a n if every A-submodule of M isfinite. This is equivalent to the condition that M has no infinite properascending chain M0 ( M1 ( M2 ( · · · ⊆M of A-submodules, orequivalently, every non-empty set of A-submodules of M has a maximalelement (with respect to the inclusion).
Emmy Noether (1882 – 1935)
The noetherian property of modules is named after Emmy Noether who is bestknown for her contributions to abstract algebra, in particular, her study of chainconditions on ideals of rings. She was a daughter of Max Noether.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Noetherian Modules
Emmy Noether
An A-module is called n o e t h e r i a n if every A-submodule of M isfinite. This is equivalent to the condition that M has no infinite properascending chain M0 ( M1 ( M2 ( · · · ⊆M of A-submodules, orequivalently, every non-empty set of A-submodules of M has a maximalelement (with respect to the inclusion).
Emmy Noether (1882 – 1935)
The noetherian property of modules is named after Emmy Noether who is bestknown for her contributions to abstract algebra, in particular, her study of chainconditions on ideals of rings. She was a daughter of Max Noether.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Noetherian Modules
Emmy Noether
An A-module is called n o e t h e r i a n if every A-submodule of M isfinite. This is equivalent to the condition that M has no infinite properascending chain M0 ( M1 ( M2 ( · · · ⊆M of A-submodules, orequivalently, every non-empty set of A-submodules of M has a maximalelement (with respect to the inclusion).
Emmy Noether (1882 – 1935)
The noetherian property of modules is named after Emmy Noether who is bestknown for her contributions to abstract algebra, in particular, her study of chainconditions on ideals of rings. She was a daughter of Max Noether.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Lasker-Noether decomposition
We will use the following Fundamental Lemma which is used inthe so-called L a s k e r - N o e t h e r D e c o m p o s i t i o n :
Lemma 4.2
Let M be a finite graded module over the standard gradedK-algebra A. Then there exist a chain of homogeneoussubmodules 0 = M0 ⊂M1 ⊂ ·· · ⊂Mr = M , and homogeneousprime ideals p1, . . . ,pr ⊆ A and integers k1, . . . ,kr withMρ/Mρ−1 ∼= (A/pρ)(−kρ), ρ = 1, . . . ,r. In particular,p1 · · ·pr M = 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Lasker-Noether decomposition
We will use the following Fundamental Lemma which is used inthe so-called L a s k e r - N o e t h e r D e c o m p o s i t i o n :
Lemma 4.2
Let M be a finite graded module over the standard gradedK-algebra A. Then there exist a chain of homogeneoussubmodules 0 = M0 ⊂M1 ⊂ ·· · ⊂Mr = M , and homogeneousprime ideals p1, . . . ,pr ⊆ A and integers k1, . . . ,kr withMρ/Mρ−1 ∼= (A/pρ)(−kρ), ρ = 1, . . . ,r. In particular,p1 · · ·pr M = 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emanuel Lasker and Max Noether
Emanuel Lasker (1868 – 1941) Max Noether (1844 – 1921)
Lasker became World Chess Champion in 1894 and held the championship until1921. He introduced the notion of a primary ideal.Max Noether was one of the leaders of nineteenth century algebraic geometry.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emanuel Lasker and Max Noether
Emanuel Lasker (1868 – 1941)
Max Noether (1844 – 1921)
Lasker became World Chess Champion in 1894 and held the championship until1921. He introduced the notion of a primary ideal.
Max Noether was one of the leaders of nineteenth century algebraic geometry.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emanuel Lasker and Max Noether
Emanuel Lasker (1868 – 1941)
Max Noether (1844 – 1921)
Lasker became World Chess Champion in 1894 and held the championship until1921. He introduced the notion of a primary ideal.
Max Noether was one of the leaders of nineteenth century algebraic geometry.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emanuel Lasker and Max Noether
Emanuel Lasker (1868 – 1941) Max Noether (1844 – 1921)
Lasker became World Chess Champion in 1894 and held the championship until1921. He introduced the notion of a primary ideal.
Max Noether was one of the leaders of nineteenth century algebraic geometry.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Emanuel Lasker and Max Noether
Emanuel Lasker (1868 – 1941) Max Noether (1844 – 1921)
Lasker became World Chess Champion in 1894 and held the championship until1921. He introduced the notion of a primary ideal.Max Noether was one of the leaders of nineteenth century algebraic geometry.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series
Let M =⊕
m∈Z Mm be a finite graded module over the standardgraded K-algebra A. Then the homogeneous components Mm,m ∈ Z, are finite dimensional K-vector spaces and Mm = 0 form << 0. Therefore, the well-known P o i n c a r é s e r i e s
PM(Z) := ∑m∈Z
(Dim K Mm )Zm
is well-defined and is a Laurent-series (with coefficients in N).
Poincaré Jules Henri (1854 – 1912)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series
Let M =⊕
m∈Z Mm be a finite graded module over the standardgraded K-algebra A. Then the homogeneous components Mm,m ∈ Z, are finite dimensional K-vector spaces and Mm = 0 form << 0.
Therefore, the well-known P o i n c a r é s e r i e sPM(Z) := ∑
m∈Z(Dim K Mm )Zm
is well-defined and is a Laurent-series (with coefficients in N).
Poincaré Jules Henri (1854 – 1912)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series
Let M =⊕
m∈Z Mm be a finite graded module over the standardgraded K-algebra A. Then the homogeneous components Mm,m ∈ Z, are finite dimensional K-vector spaces and Mm = 0 form << 0. Therefore, the well-known P o i n c a r é s e r i e s
PM(Z) := ∑m∈Z
(Dim K Mm )Zm
is well-defined and is a Laurent-series (with coefficients in N).
Poincaré Jules Henri (1854 – 1912)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series
Let M =⊕
m∈Z Mm be a finite graded module over the standardgraded K-algebra A. Then the homogeneous components Mm,m ∈ Z, are finite dimensional K-vector spaces and Mm = 0 form << 0. Therefore, the well-known P o i n c a r é s e r i e s
PM(Z) := ∑m∈Z
(Dim K Mm )Zm
is well-defined and is a Laurent-series (with coefficients in N).
Poincaré Jules Henri (1854 – 1912)
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd...
The following general fundamental lemma was already in thework of Hilbert with a complicated proof :
Lemma 4.3
Let M be a finite graded module over the standardly gradedK-algebra A = K[t0, . . . , tn], t0, . . . , tn ∈ A1. Then
PM =R
(1−Z)n+1 with a Laurent-polynomial R ∈ Z[Z±1].
If M 6= 0, then after canceling the highest possible power of(1−Z), we get a unique representation :
PM =Q
(1−Z)d+1 , d ≥−1
with a Laurent-polynomial Q ∈ Z[Z±1], Q(1) 6= 0.
For M = 0, d =−1 and Q = 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd...
The following general fundamental lemma was already in thework of Hilbert with a complicated proof :
Lemma 4.3
Let M be a finite graded module over the standardly gradedK-algebra A = K[t0, . . . , tn], t0, . . . , tn ∈ A1. Then
PM =R
(1−Z)n+1 with a Laurent-polynomial R ∈ Z[Z±1].
If M 6= 0, then after canceling the highest possible power of(1−Z), we get a unique representation :
PM =Q
(1−Z)d+1 , d ≥−1
with a Laurent-polynomial Q ∈ Z[Z±1], Q(1) 6= 0.
For M = 0, d =−1 and Q = 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd...
The following general fundamental lemma was already in thework of Hilbert with a complicated proof :
Lemma 4.3
Let M be a finite graded module over the standardly gradedK-algebra A = K[t0, . . . , tn], t0, . . . , tn ∈ A1. Then
PM =R
(1−Z)n+1 with a Laurent-polynomial R ∈ Z[Z±1].
If M 6= 0, then after canceling the highest possible power of(1−Z), we get a unique representation :
PM =Q
(1−Z)d+1 , d ≥−1
with a Laurent-polynomial Q ∈ Z[Z±1], Q(1) 6= 0.
For M = 0, d =−1 and Q = 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd...
The following general fundamental lemma was already in thework of Hilbert with a complicated proof :
Lemma 4.3
Let M be a finite graded module over the standardly gradedK-algebra A = K[t0, . . . , tn], t0, . . . , tn ∈ A1. Then
PM =R
(1−Z)n+1 with a Laurent-polynomial R ∈ Z[Z±1].
If M 6= 0, then after canceling the highest possible power of(1−Z), we get a unique representation :
PM =Q
(1−Z)d+1 , d ≥−1
with a Laurent-polynomial Q ∈ Z[Z±1], Q(1) 6= 0.
For M = 0, d =−1 and Q = 0.Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd. ...
The partial fraction decomposition is
PM = Q̃+d
∑i=0
ci
(1−Z)i+1 ≡d
∑i=0
ci
(1−Z)i+1
with a uniquely determined Laurent-polynomial Q̃ ∈ Z[Z±1] anduniquely determined integers c0, . . . ,cd ∈ Z, where we writeG≡ H for two Laurent-series G, H if and only if they differ by aLaurent-polynomial.
Using the formula (1−Z)−(n+1) = ∑m(m+n
n
)Zm which can be proved
directly by differentaiting (termwise) n-times the geometric series(1−Z)−1 = ∑m Zm , we get :
For m >> 0 (more precisely for m > deg Q̃), we have
Dim K Mm = χM(m) :=d
∑i=0
ci
(m+ i
i
)for m >> 0 ,
where χM is a polynomial function of degree d.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd. ...
The partial fraction decomposition is
PM = Q̃+d
∑i=0
ci
(1−Z)i+1 ≡d
∑i=0
ci
(1−Z)i+1
with a uniquely determined Laurent-polynomial Q̃ ∈ Z[Z±1] anduniquely determined integers c0, . . . ,cd ∈ Z, where we writeG≡ H for two Laurent-series G, H if and only if they differ by aLaurent-polynomial.Using the formula (1−Z)−(n+1) = ∑m
(m+nn
)Zm which can be proved
directly by differentaiting (termwise) n-times the geometric series(1−Z)−1 = ∑m Zm , we get :
For m >> 0 (more precisely for m > deg Q̃), we have
Dim K Mm = χM(m) :=d
∑i=0
ci
(m+ i
i
)for m >> 0 ,
where χM is a polynomial function of degree d.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Poincaré series Contd. ...
The partial fraction decomposition is
PM = Q̃+d
∑i=0
ci
(1−Z)i+1 ≡d
∑i=0
ci
(1−Z)i+1
with a uniquely determined Laurent-polynomial Q̃ ∈ Z[Z±1] anduniquely determined integers c0, . . . ,cd ∈ Z, where we writeG≡ H for two Laurent-series G, H if and only if they differ by aLaurent-polynomial.Using the formula (1−Z)−(n+1) = ∑m
(m+nn
)Zm which can be proved
directly by differentaiting (termwise) n-times the geometric series(1−Z)−1 = ∑m Zm , we get :
For m >> 0 (more precisely for m > deg Q̃), we have
Dim K Mm = χM(m) :=d
∑i=0
ci
(m+ i
i
)for m >> 0 ,
where χM is a polynomial function of degree d.Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
In particular, if d ≥ 0, then
Dim K Mm = χM(m)∼ cd ·md
d!= O(md) for m→ ∞ .
where ∼ denote the asymptotic equality and where O is the “BigO” symbol.
The symbol “Big O” is first introduced by the number theorist Paul Bachmann(1837-1920) in 1894. Another number theorist Edmund Landau (1877-1938)adopted it and was inspired to introduce the “small o” notation in 1909. Thesesymbols describe the limiting behaviour of a function. More precisely : ForR-valued functions f , g : U→ R defined on some subset U ⊆ R, one writes :(i) f (x) = O(g(x)) as x→ ∞ (|f | is bounded above by |g|, up to constant factor,asymptotically) if there exists a constant M > 0 and a real number x0 ∈ R suchthat |f (x)| ≤M |g(x)| for all x≥ x0, or equivalently limsupx→∞ |f (x)/g(x)|< ∞.(ii) f (x) = o(g(x)) as x→ ∞ (f is dominated by g asymptotically) iflimx→∞ |f (x)/g(x)|= 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
In particular, if d ≥ 0, then
Dim K Mm = χM(m)∼ cd ·md
d!= O(md) for m→ ∞ .
where ∼ denote the asymptotic equality and where O is the “BigO” symbol.
The symbol “Big O” is first introduced by the number theorist Paul Bachmann(1837-1920) in 1894. Another number theorist Edmund Landau (1877-1938)adopted it and was inspired to introduce the “small o” notation in 1909. Thesesymbols describe the limiting behaviour of a function. More precisely : ForR-valued functions f , g : U→ R defined on some subset U ⊆ R, one writes :(i) f (x) = O(g(x)) as x→ ∞ (|f | is bounded above by |g|, up to constant factor,asymptotically) if there exists a constant M > 0 and a real number x0 ∈ R suchthat |f (x)| ≤M |g(x)| for all x≥ x0, or equivalently limsupx→∞ |f (x)/g(x)|< ∞.(ii) f (x) = o(g(x)) as x→ ∞ (f is dominated by g asymptotically) iflimx→∞ |f (x)/g(x)|= 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
In particular, if d ≥ 0, then
Dim K Mm = χM(m)∼ cd ·md
d!= O(md) for m→ ∞ .
where ∼ denote the asymptotic equality and where O is the “BigO” symbol.
The symbol “Big O” is first introduced by the number theorist Paul Bachmann(1837-1920) in 1894. Another number theorist Edmund Landau (1877-1938)adopted it and was inspired to introduce the “small o” notation in 1909. Thesesymbols describe the limiting behaviour of a function. More precisely : ForR-valued functions f , g : U→ R defined on some subset U ⊆ R, one writes :(i) f (x) = O(g(x)) as x→ ∞ (|f | is bounded above by |g|, up to constant factor,asymptotically) if there exists a constant M > 0 and a real number x0 ∈ R suchthat |f (x)| ≤M |g(x)| for all x≥ x0, or equivalently limsupx→∞ |f (x)/g(x)|< ∞.(ii) f (x) = o(g(x)) as x→ ∞ (f is dominated by g asymptotically) iflimx→∞ |f (x)/g(x)|= 0.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert series and Hilbert polynomial
The case d =−1 is characterised by Dim K Mm = 0 for m >> 0, orby Dim K M = ∑m∈Z Dim K Mm = Q(1)< ∞.
Incidently, instead of Poincaré-series it is comfortable to considerthe H i l b e r t - s e r i e s
HM = ∑m∈Z hM(m)Zm = P/(1−Z)≡∑d+1i=0 ei/(1−Z)i+1
with the Hilbert function hM : Z−→ N :
hM(m) = ∑k≤m Dim K Mm = Dim K(⊕
k≤mMk)
and put ei := ci−1, if i > 0, and e0 := Q̃(1).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert-series
For large m >> 0, the values hM(m) are equal to the values of theHilbert-Polynomial
HM(m) = ∑d+1i=0 ei
(m+ i
i
)∼ ed+1 ·md+1/(d+1)= O(md+1) .
The integer d is an approximate measure of the size of M and iscalled the p r o j e c t i v e d i m e n s i o n pd(M) and d+1 is the(a f f i n e or K r u l l-) d i m e n s i o n d(M) of the graded moduleM.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert-series
For large m >> 0, the values hM(m) are equal to the values of theHilbert-Polynomial
HM(m) = ∑d+1i=0 ei
(m+ i
i
)∼ ed+1 ·md+1/(d+1)= O(md+1) .
The integer d is an approximate measure of the size of M and iscalled the p r o j e c t i v e d i m e n s i o n pd(M) and d+1 is the(a f f i n e or K r u l l-) d i m e n s i o n d(M) of the graded moduleM.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert-series
For large m >> 0, the values hM(m) are equal to the values of theHilbert-Polynomial
HM(m) = ∑d+1i=0 ei
(m+ i
i
)∼ ed+1 ·md+1/(d+1)= O(md+1) .
The integer d is an approximate measure of the size of M and iscalled the p r o j e c t i v e d i m e n s i o n pd(M) and d+1 is the(a f f i n e or K r u l l-) d i m e n s i o n d(M) of the graded moduleM.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective dimension and Multiplicity.
The integer d is called the p r o j e c t i v e d i m e n s i o n pd(M)and d+1 is the (a f f i n e or K r u l l-) d i m e n s i o n d(M) of thegraded module M.
The integer e(M) := ed+1 = ed(M)(= cpd(M) if if pd(M)≥ 0) iscalled the m u l t i p l i c i t y o f t h e g r a d e d m o d u l e M ifpd(M)≥ 0.
Note that e(M) is positive if M 6= 0. If M = 0, then d(0) = e(0) = 0.
If PM =Q
(1−Z)pd(M), then HM =
Q(1−Z)1+pd(M)
and e(M) = Q(1) ,
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective dimension and Multiplicity.
The integer d is called the p r o j e c t i v e d i m e n s i o n pd(M)and d+1 is the (a f f i n e or K r u l l-) d i m e n s i o n d(M) of thegraded module M.
The integer e(M) := ed+1 = ed(M)(= cpd(M) if if pd(M)≥ 0) iscalled the m u l t i p l i c i t y o f t h e g r a d e d m o d u l e M ifpd(M)≥ 0.
Note that e(M) is positive if M 6= 0. If M = 0, then d(0) = e(0) = 0.
If PM =Q
(1−Z)pd(M), then HM =
Q(1−Z)1+pd(M)
and e(M) = Q(1) ,
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective dimension and Multiplicity.
The integer d is called the p r o j e c t i v e d i m e n s i o n pd(M)and d+1 is the (a f f i n e or K r u l l-) d i m e n s i o n d(M) of thegraded module M.
The integer e(M) := ed+1 = ed(M)(= cpd(M) if if pd(M)≥ 0) iscalled the m u l t i p l i c i t y o f t h e g r a d e d m o d u l e M ifpd(M)≥ 0.
Note that e(M) is positive if M 6= 0. If M = 0, then d(0) = e(0) = 0.
If PM =Q
(1−Z)pd(M), then HM =
Q(1−Z)1+pd(M)
and e(M) = Q(1) ,
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective dimension and Multiplicity.
The integer d is called the p r o j e c t i v e d i m e n s i o n pd(M)and d+1 is the (a f f i n e or K r u l l-) d i m e n s i o n d(M) of thegraded module M.
The integer e(M) := ed+1 = ed(M)(= cpd(M) if if pd(M)≥ 0) iscalled the m u l t i p l i c i t y o f t h e g r a d e d m o d u l e M ifpd(M)≥ 0.
Note that e(M) is positive if M 6= 0. If M = 0, then d(0) = e(0) = 0.
If PM =Q
(1−Z)pd(M), then HM =
Q(1−Z)1+pd(M)
and e(M) = Q(1) ,
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Example
Example 4.4
If M = A = P = K[T0, . . . ,Tn], n ∈ N, then
PK[T0,...,Tn]=1/(1−Z)n and HK[T0,...,Tn]=1/(1−Z)n+1 .
In particular, the projective dimension pd(K[T0, . . . ,Tn]) = n, theaffine dimension d(K[T0, . . . ,Tn]) = n+1 and the multiplicitye(K[T0, . . . ,Tn]) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Example
Example 4.4
If M = A = P = K[T0, . . . ,Tn], n ∈ N, then
PK[T0,...,Tn]=1/(1−Z)n and HK[T0,...,Tn]=1/(1−Z)n+1 .
In particular, the projective dimension pd(K[T0, . . . ,Tn]) = n, theaffine dimension d(K[T0, . . . ,Tn]) = n+1 and the multiplicitye(K[T0, . . . ,Tn]) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.
Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules for Dimension and Multiplicity
The following computational rules for d(M) and e(M) are easy toverify :
Computational Rules 4.5Let K be a field and A =
⊕n∈N An be standard graded K-algebra. Then
for a finitely generated graded module M over A, we have
(1) d(M) = d(M(−k)) and e(M) = e(M(−k)), k ∈ Z.
(2) Let 0→Mr→Mr−1→ ··· →M0→ 0 is an exact sequence ofhomogeneous homomorphisms and d := max
1≤ρ≤r{d(Mρ)}, then
∑ρ ,d(Mρ )=d
(−1)ρ e(Mρ) = 0 .
(3) Let f ∈ Aδ be a homogeneous element of degree δ > 0. Thend(M/fM)≥ d(M)−1.Moreover, if f is a non-zero divisor for M and M 6= 0, thend(M/fM)= d(M)−1 and e(M/fM)=δ · e(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules Contd ...
Computational Rules 4.6
(4) Let 0 = M0 ⊆M1 ⊆ ...⊆Mr = M be a chain of gradedA-submodules of M. Then ( A s s o c i a t i v i t y F o r n u l a e ) :
d:=d(M)=max1≤ρ≤r
{d(Mρ/Mρ−1)} and e(M)= ∑ρ ,d(Mρ/Mρ−1)=d
e(Mρ/Mρ−1).
(5) Moreover, if in (4) if there are homogeneous prime ideals p1, . . . ,prand integers k1, . . . ,kr with Mρ/Mρ−1 ∼= (A/pρ)(−kρ),ρ = 1, . . . ,r, then
d(M) = max1≤ρ≤r
{d(A/pρ)} , and e(M) = ∑ρ ,d(A/pρ )=d(M)
e(A/pρ) .
In particular, if M 6= 0, then there are prime ideals pρ withd(A/pρ) = d(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules Contd ...
Computational Rules 4.6
(4) Let 0 = M0 ⊆M1 ⊆ ...⊆Mr = M be a chain of gradedA-submodules of M. Then ( A s s o c i a t i v i t y F o r n u l a e ) :
d:=d(M)=max1≤ρ≤r
{d(Mρ/Mρ−1)} and e(M)= ∑ρ ,d(Mρ/Mρ−1)=d
e(Mρ/Mρ−1).
(5) Moreover, if in (4) if there are homogeneous prime ideals p1, . . . ,prand integers k1, . . . ,kr with Mρ/Mρ−1 ∼= (A/pρ)(−kρ),ρ = 1, . . . ,r, then
d(M) = max1≤ρ≤r
{d(A/pρ)} , and e(M) = ∑ρ ,d(A/pρ )=d(M)
e(A/pρ) .
In particular, if M 6= 0, then there are prime ideals pρ withd(A/pρ) = d(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Computational rules Contd ...
Computational Rules 4.6
(4) Let 0 = M0 ⊆M1 ⊆ ...⊆Mr = M be a chain of gradedA-submodules of M. Then ( A s s o c i a t i v i t y F o r n u l a e ) :
d:=d(M)=max1≤ρ≤r
{d(Mρ/Mρ−1)} and e(M)= ∑ρ ,d(Mρ/Mρ−1)=d
e(Mρ/Mρ−1).
(5) Moreover, if in (4) if there are homogeneous prime ideals p1, . . . ,prand integers k1, . . . ,kr with Mρ/Mρ−1 ∼= (A/pρ)(−kρ),ρ = 1, . . . ,r, then
d(M) = max1≤ρ≤r
{d(A/pρ)} , and e(M) = ∑ρ ,d(A/pρ )=d(M)
e(A/pρ) .
In particular, if M 6= 0, then there are prime ideals pρ withd(A/pρ) = d(M) .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§5 Projective Nullstellensatz
Let K be a field. First, for a standard graded K-algebra
A =⊕m∈N
Am = K[t0, ..., tn] , t0, ..., tn ∈ A1
we define the p r o j e c t i v e a l g e b r a i c s e t
PA(K)
o f K - v a l u e d p o i n t s.
For the polynomial algebra A = P = K[T0, . . . ,Tn], this is
PP(K) := Pn(K)= {〈τ〉= 〈τ0, . . . ,τn〉 | τ = (τ0, . . . ,τn) ∈ Kn+1 \{0}}
the n-dimensional projective space over K
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§5 Projective Nullstellensatz
Let K be a field. First, for a standard graded K-algebra
A =⊕m∈N
Am = K[t0, ..., tn] , t0, ..., tn ∈ A1
we define the p r o j e c t i v e a l g e b r a i c s e t
PA(K)
o f K - v a l u e d p o i n t s.
For the polynomial algebra A = P = K[T0, . . . ,Tn], this is
PP(K) := Pn(K)= {〈τ〉= 〈τ0, . . . ,τn〉 | τ = (τ0, . . . ,τn) ∈ Kn+1 \{0}}
the n-dimensional projective space over K
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§5 Projective Nullstellensatz
Let K be a field. First, for a standard graded K-algebra
A =⊕m∈N
Am = K[t0, ..., tn] , t0, ..., tn ∈ A1
we define the p r o j e c t i v e a l g e b r a i c s e t
PA(K)
o f K - v a l u e d p o i n t s.
For the polynomial algebra A = P = K[T0, . . . ,Tn], this is
PP(K) := Pn(K)= {〈τ〉= 〈τ0, . . . ,τn〉 | τ = (τ0, . . . ,τn) ∈ Kn+1 \{0}}
the n-dimensional projective space over K
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets
Let A be the kernel of the substitution homomorphismε : K[T0, ...,Tn]→ A , Ti 7→ ti , i = 0, ...,n . Then A∼= P/A and bydefinition :
PK(A) = V+(A) := {〈τ〉 ∈ Pn(K) |F(τ) = 0 for all
homogeneous F ∈ A} ⊆ Pn(K)
set of common zeroes of the homogeneous relation ideal A.
Further, if F1, . . . ,Fm ∈ A is a homogeneous system of generatorsfor A, then
PA(K)=V+(F1, . . . ,Fm)={〈τ〉 ∈ Pn(K) | Fi(τ)=0 , i=1, . . . ,m}
It is easy to see that the description of PA(K) is independent ofthe representation A∼= P/A.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets
Let A be the kernel of the substitution homomorphismε : K[T0, ...,Tn]→ A , Ti 7→ ti , i = 0, ...,n . Then A∼= P/A and bydefinition :
PK(A) = V+(A) := {〈τ〉 ∈ Pn(K) |F(τ) = 0 for all
homogeneous F ∈ A} ⊆ Pn(K)
set of common zeroes of the homogeneous relation ideal A.
Further, if F1, . . . ,Fm ∈ A is a homogeneous system of generatorsfor A, then
PA(K)=V+(F1, . . . ,Fm)={〈τ〉 ∈ Pn(K) | Fi(τ)=0 , i=1, . . . ,m}
It is easy to see that the description of PA(K) is independent ofthe representation A∼= P/A.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets
Let A be the kernel of the substitution homomorphismε : K[T0, ...,Tn]→ A , Ti 7→ ti , i = 0, ...,n . Then A∼= P/A and bydefinition :
PK(A) = V+(A) := {〈τ〉 ∈ Pn(K) |F(τ) = 0 for all
homogeneous F ∈ A} ⊆ Pn(K)
set of common zeroes of the homogeneous relation ideal A.
Further, if F1, . . . ,Fm ∈ A is a homogeneous system of generatorsfor A, then
PA(K)=V+(F1, . . . ,Fm)={〈τ〉 ∈ Pn(K) | Fi(τ)=0 , i=1, . . . ,m}
It is easy to see that the description of PA(K) is independent ofthe representation A∼= P/A.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets Contd. ...
If f ∈ A is a homogeneous element with a homogeneousrepresentative F ∈ P, then the zero set
V+(f ) = {〈τ〉 ∈ PA(K) | F(τ) = 0}
of f in PA(K) is well-defined.
In particular, for a homogeneous ideal a⊆ A, the representation
PA/a(K) = V+(f1, . . . , fr) =r⋂
i=1
V+(fρ)⊆ PA(K)
where f1, . . . , fr is a homogeneous system of generators for a.
A point 〈τ〉 ∈Pn(K) defines the homogeneous prime ideal P〈τ〉 ⊆ P generatedby the homogeneous polynomials F ∈ P which vanish on τ . It is alreadygenerated by the liner forms F ∈ P1 and P/P〈τ〉 ∼= K[T] .
In particular, the projective dimension d(P/P〈τ〉) = 0 and the multiplicitye(P/P〈τ〉) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets Contd. ...
If f ∈ A is a homogeneous element with a homogeneousrepresentative F ∈ P, then the zero set
V+(f ) = {〈τ〉 ∈ PA(K) | F(τ) = 0}
of f in PA(K) is well-defined.
In particular, for a homogeneous ideal a⊆ A, the representation
PA/a(K) = V+(f1, . . . , fr) =r⋂
i=1
V+(fρ)⊆ PA(K)
where f1, . . . , fr is a homogeneous system of generators for a.
A point 〈τ〉 ∈Pn(K) defines the homogeneous prime ideal P〈τ〉 ⊆ P generatedby the homogeneous polynomials F ∈ P which vanish on τ . It is alreadygenerated by the liner forms F ∈ P1 and P/P〈τ〉 ∼= K[T] .
In particular, the projective dimension d(P/P〈τ〉) = 0 and the multiplicitye(P/P〈τ〉) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets Contd. ...
If f ∈ A is a homogeneous element with a homogeneousrepresentative F ∈ P, then the zero set
V+(f ) = {〈τ〉 ∈ PA(K) | F(τ) = 0}
of f in PA(K) is well-defined.
In particular, for a homogeneous ideal a⊆ A, the representation
PA/a(K) = V+(f1, . . . , fr) =r⋂
i=1
V+(fρ)⊆ PA(K)
where f1, . . . , fr is a homogeneous system of generators for a.
A point 〈τ〉 ∈Pn(K) defines the homogeneous prime ideal P〈τ〉 ⊆ P generatedby the homogeneous polynomials F ∈ P which vanish on τ . It is alreadygenerated by the liner forms F ∈ P1 and P/P〈τ〉 ∼= K[T] .
In particular, the projective dimension d(P/P〈τ〉) = 0 and the multiplicitye(P/P〈τ〉) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets Contd. ...
If f ∈ A is a homogeneous element with a homogeneousrepresentative F ∈ P, then the zero set
V+(f ) = {〈τ〉 ∈ PA(K) | F(τ) = 0}
of f in PA(K) is well-defined.
In particular, for a homogeneous ideal a⊆ A, the representation
PA/a(K) = V+(f1, . . . , fr) =r⋂
i=1
V+(fρ)⊆ PA(K)
where f1, . . . , fr is a homogeneous system of generators for a.
A point 〈τ〉 ∈Pn(K) defines the homogeneous prime ideal P〈τ〉 ⊆ P generatedby the homogeneous polynomials F ∈ P which vanish on τ . It is alreadygenerated by the liner forms F ∈ P1 and P/P〈τ〉 ∼= K[T] .
In particular, the projective dimension d(P/P〈τ〉) = 0 and the multiplicitye(P/P〈τ〉) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Projective algebraic sets Contd. ...
If f ∈ A is a homogeneous element with a homogeneousrepresentative F ∈ P, then the zero set
V+(f ) = {〈τ〉 ∈ PA(K) | F(τ) = 0}
of f in PA(K) is well-defined.
In particular, for a homogeneous ideal a⊆ A, the representation
PA/a(K) = V+(f1, . . . , fr) =r⋂
i=1
V+(fρ)⊆ PA(K)
where f1, . . . , fr is a homogeneous system of generators for a.
A point 〈τ〉 ∈Pn(K) defines the homogeneous prime ideal P〈τ〉 ⊆ P generatedby the homogeneous polynomials F ∈ P which vanish on τ . It is alreadygenerated by the liner forms F ∈ P1 and P/P〈τ〉 ∼= K[T] .
In particular, the projective dimension d(P/P〈τ〉) = 0 and the multiplicitye(P/P〈τ〉) = 1.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).
( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.
This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Points in Projective space
Conversely, if P⊆ P is a homogeneous prime ideal with d(P/P) = 0 ande(P/P) = 1, then P=P〈τ〉 for a unique point 〈τ〉 ∈Pn(K).( Proof The K-subspace P1 ⊆ P1 is of codimension 1, sincee(P/P)≥ DimK(P/P)m for every m ∈ N and every prime ideal P⊆ P withd(P/P) = 0. )
Further, 〈τ〉 ∈ PA(K) if and only if A= Ker ε ⊆P〈τ〉.This proves :
Lemma 5.1
The points in PA(K) corresponds to unique homogeneous primeideals p⊆ A with d(A/p) = 0 and e(A/p) = 1.
Lemma 5.2
The multiplicity of a standard graded K-algebra which is adomain and of the projective dimension 0 is equal to the degree ofa finite field extension of K.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert’s Nullstellensatz
We assume now on that A is a standard graded K-algebra ofprojective dimension d = pd(A)≥ 0 and hence of (affine Krull-)dimension d(A) = d+1≥ 1.
Now first we note the classical Hilbert’s Nullstellensatz :
Theorem 5.3 ( H i l b e r t ’ s N u l l s t e l l e n s a t z )
Let K be an algebraically closed field and let f1, . . . , fr ∈ A behomogeneous elements of positive degrees, r ≤ d. Then f1, . . . , frhave a common zero in PA(K), i. e.,/0 6= PA/a(K) = V+(f1, . . . , fr)⊆ PA(K), a := Af1 + · · ·+Afr .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert’s Nullstellensatz
We assume now on that A is a standard graded K-algebra ofprojective dimension d = pd(A)≥ 0 and hence of (affine Krull-)dimension d(A) = d+1≥ 1.
Now first we note the classical Hilbert’s Nullstellensatz :
Theorem 5.3 ( H i l b e r t ’ s N u l l s t e l l e n s a t z )
Let K be an algebraically closed field and let f1, . . . , fr ∈ A behomogeneous elements of positive degrees, r ≤ d. Then f1, . . . , frhave a common zero in PA(K), i. e.,/0 6= PA/a(K) = V+(f1, . . . , fr)⊆ PA(K), a := Af1 + · · ·+Afr .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Hilbert’s Nullstellensatz
We assume now on that A is a standard graded K-algebra ofprojective dimension d = pd(A)≥ 0 and hence of (affine Krull-)dimension d(A) = d+1≥ 1.
Now first we note the classical Hilbert’s Nullstellensatz :
Theorem 5.3 ( H i l b e r t ’ s N u l l s t e l l e n s a t z )
Let K be an algebraically closed field and let f1, . . . , fr ∈ A behomogeneous elements of positive degrees, r ≤ d. Then f1, . . . , frhave a common zero in PA(K), i. e.,/0 6= PA/a(K) = V+(f1, . . . , fr)⊆ PA(K), a := Af1 + · · ·+Afr .
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Proof of Hilbert’s Nullstellensatz
Proof (Induction on d and r.) If d = 0, then r = 0. By Lemma 4.2 there exists ahomogeneous prime ideal p⊆ A with pd(A/p) = 0. By Lemma 5.2, necessarilye(A/p) = 1 and p definies — by Lemma 5.2 — a point in PA(K).For the inductive step from d to d+1, consider a prime ideal p⊆ A withd = pd(A/p). It is enought to prove that/0 6= V+(f 1, . . . , f r)⊆ PA/p(K)⊆ PA(K), where f 1, . . . , f r denote the residueclasses of f1, . . . , fr in A/p. We may therefore assume that A is an integraldomain and fr 6= 0. Then pd(A/Afr) = d−1. By induction hypothesis it follows/0 6= V+(f 1, . . . , f r−1) = V+(f1, . . . , fr)⊆ PA/Afr (K), where now f 1, . . . , f r−1 arethe residue classes in A/Afr.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Identity theorem
The following theorem is known as the Identity theorem forpolynomial functions.
Theorem 5.4 ( I d e n t i t y t h e o r e m )
Let K be an algebraically closed field and let A be domain. If ahomogeneous element f ∈ A vanish at all points of PA(K), thenf = 0.
Proof (Induction on d.) Let d > 0 and deg f > 0. By induction hypothesis istf ∈ p for all homogeneous prime ideal p 6= 0 in A. If f 6= 0, then there existhomogeneous prime ideal q 6= 0 with f 6∈ q. Further, M := A/Af has theprojective dimension d−1. If 0 = M0 ⊂M1 ⊂ ·· · ⊂Mr = M is a chain ofsubmodules with Mρ/Mρ−1 = (A/pρ )(−kρ ) as in Lemma 4.2, then f ∈ pρ andthe prime ideals p⊆ A with pd(A/p) := d−1 = pd(M) and f ∈ p belongs to thefinitely many prime ideal p1, . . . ,pr. It follows that there exists a g ∈ A1 which isnot in the union of these ideals (since p∩A1 ⊂ A1, if pd(A/p)≥ 0 ist). If q⊆ Ais a homogeneous prime ideal of the projective dimension d−1 with g ∈ q, thenf 6∈ q.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§6 Analogs of HNS to 2-fields
The analogous theorems for 2-fields are :
Theorem 6.1
Let K be a 2-field, A be of odd multiplicity and let f1, . . . , fr ∈ A behomogeneous elements of positive odd degrees, r ≤ d. Thenf1, . . . , fr have a common zero in PA(K), i. e.,/0 6= PA/a(K) = V+(f1, . . . , fr)⊆ PA(K), a := Af1 + · · ·+Afr.
Proof For M := A, let 0 = M0 ⊂M1 ⊂ ·· · ⊂Mr = M be a chain withMρ/Mρ−1 = (A/pρ )(−kρ ) as in Lemma 4.2. Thene(A) = ∑ρ ,pd(A/pρ )=d e(A/pρ ). It follows : If e (A) is odd, then at least one ofe (A/pρ ) with pd(A/pρ ) = d is also odd.If d = 0, then by Lemma 5.1 and Lemma 3.2 necessarily e(A/pρ ) = 1 for onepρ with pd(A/pρ ) = 0, and such a prime ideal pρ defines a point in PA(K). Forthe inductive step from d to d+1, we may assume that A is an integral domainand fr 6= 0. Then e(A/Afr) = e(A) ·deg fr is also odd and pd(A/Afr) = d, and byapplying the induction hypothesis to A/Afr and the residue classes f 1, . . . , f r−1,the assertion follows.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
§6 Analogs of HNS to 2-fields
The analogous theorems for 2-fields are :
Theorem 6.1
Let K be a 2-field, A be of odd multiplicity and let f1, . . . , fr ∈ A behomogeneous elements of positive odd degrees, r ≤ d. Thenf1, . . . , fr have a common zero in PA(K), i. e.,/0 6= PA/a(K) = V+(f1, . . . , fr)⊆ PA(K), a := Af1 + · · ·+Afr.
Proof For M := A, let 0 = M0 ⊂M1 ⊂ ·· · ⊂Mr = M be a chain withMρ/Mρ−1 = (A/pρ )(−kρ ) as in Lemma 4.2. Thene(A) = ∑ρ ,pd(A/pρ )=d e(A/pρ ). It follows : If e (A) is odd, then at least one ofe (A/pρ ) with pd(A/pρ ) = d is also odd.If d = 0, then by Lemma 5.1 and Lemma 3.2 necessarily e(A/pρ ) = 1 for onepρ with pd(A/pρ ) = 0, and such a prime ideal pρ defines a point in PA(K). Forthe inductive step from d to d+1, we may assume that A is an integral domainand fr 6= 0. Then e(A/Afr) = e(A) ·deg fr is also odd and pd(A/Afr) = d, and byapplying the induction hypothesis to A/Afr and the residue classes f 1, . . . , f r−1,the assertion follows.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Theorem 6.2
Let K be a 2-field and let A be domain of odd multiplicity. If ahomogeneous element f ∈ A vanish at all points of PA(K), thenf = 0.
Proof As in the proof of Theorem 5.4, let d > 0 and deg f > 0. By theinduction hypothesis, f belongs to the intersection of all homogeneousprime ideals p 6= 0, with e(A/p) is odd. If f 6= 0, then there exists aprime ideal q with f 6∈ q and e(A/q) odd, since as in the proof of theTheorem 5.4 one can construct ideal q for a chosen g ∈ A1, g 6= 0 withe(A/Ag) = e(A).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Real Analytic Manifolds
We note the following Corollary to Theorem 6.2 for K = R.
The projective space PP(R) = Pn(R) is an n-dimensional real-analyticmanifold. A projective algebraic set PP/A(R)⊆ Pn(R) is a closedsubset.
Corollary 6.3If the standard graded R-Algebra A := R[T0, . . . ,Tn]/P is an integraldomain with d := pd(A)≥ 0 and odd multiplicity, then there exists annon-empty open subset in PA(R) = V+(P)⊆ Pn(R) which is ad-dimensional analytic submanifold in Pn(R).
Proof Consider the algebraic Kähler differential forms on theprojective variety corresponding to A. By 6.2 there exist smooth pointsin the variety PA(R)., i. e., this variety is a real analytic submanifold ofdimension d.For details, see R. HARTSHORNE, Algebraic Geometry, II,8,in particular, Corollary8.16. — However, the set of smooth points ofdimension d is, not in general, dense in PA(R).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Real Analytic Manifolds
We note the following Corollary to Theorem 6.2 for K = R.The projective space PP(R) = Pn(R) is an n-dimensional real-analyticmanifold. A projective algebraic set PP/A(R)⊆ Pn(R) is a closedsubset.
Corollary 6.3If the standard graded R-Algebra A := R[T0, . . . ,Tn]/P is an integraldomain with d := pd(A)≥ 0 and odd multiplicity, then there exists annon-empty open subset in PA(R) = V+(P)⊆ Pn(R) which is ad-dimensional analytic submanifold in Pn(R).
Proof Consider the algebraic Kähler differential forms on theprojective variety corresponding to A. By 6.2 there exist smooth pointsin the variety PA(R)., i. e., this variety is a real analytic submanifold ofdimension d.For details, see R. HARTSHORNE, Algebraic Geometry, II,8,in particular, Corollary8.16. — However, the set of smooth points ofdimension d is, not in general, dense in PA(R).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Real Analytic Manifolds
We note the following Corollary to Theorem 6.2 for K = R.The projective space PP(R) = Pn(R) is an n-dimensional real-analyticmanifold. A projective algebraic set PP/A(R)⊆ Pn(R) is a closedsubset.
Corollary 6.3If the standard graded R-Algebra A := R[T0, . . . ,Tn]/P is an integraldomain with d := pd(A)≥ 0 and odd multiplicity, then there exists annon-empty open subset in PA(R) = V+(P)⊆ Pn(R) which is ad-dimensional analytic submanifold in Pn(R).
Proof Consider the algebraic Kähler differential forms on theprojective variety corresponding to A. By 6.2 there exist smooth pointsin the variety PA(R)., i. e., this variety is a real analytic submanifold ofdimension d.For details, see R. HARTSHORNE, Algebraic Geometry, II,8,in particular, Corollary8.16. — However, the set of smooth points ofdimension d is, not in general, dense in PA(R).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Real Analytic Manifolds
We note the following Corollary to Theorem 6.2 for K = R.The projective space PP(R) = Pn(R) is an n-dimensional real-analyticmanifold. A projective algebraic set PP/A(R)⊆ Pn(R) is a closedsubset.
Corollary 6.3If the standard graded R-Algebra A := R[T0, . . . ,Tn]/P is an integraldomain with d := pd(A)≥ 0 and odd multiplicity, then there exists annon-empty open subset in PA(R) = V+(P)⊆ Pn(R) which is ad-dimensional analytic submanifold in Pn(R).
Proof Consider the algebraic Kähler differential forms on theprojective variety corresponding to A. By 6.2 there exist smooth pointsin the variety PA(R)., i. e., this variety is a real analytic submanifold ofdimension d.
For details, see R. HARTSHORNE, Algebraic Geometry, II,8,in particular, Corollary8.16. — However, the set of smooth points ofdimension d is, not in general, dense in PA(R).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
Real Analytic Manifolds
We note the following Corollary to Theorem 6.2 for K = R.The projective space PP(R) = Pn(R) is an n-dimensional real-analyticmanifold. A projective algebraic set PP/A(R)⊆ Pn(R) is a closedsubset.
Corollary 6.3If the standard graded R-Algebra A := R[T0, . . . ,Tn]/P is an integraldomain with d := pd(A)≥ 0 and odd multiplicity, then there exists annon-empty open subset in PA(R) = V+(P)⊆ Pn(R) which is ad-dimensional analytic submanifold in Pn(R).
Proof Consider the algebraic Kähler differential forms on theprojective variety corresponding to A. By 6.2 there exist smooth pointsin the variety PA(R)., i. e., this variety is a real analytic submanifold ofdimension d.For details, see R. HARTSHORNE, Algebraic Geometry, II,8,in particular, Corollary8.16. — However, the set of smooth points ofdimension d is, not in general, dense in PA(R).
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz
Borsuk-UlamTheorem and
Hilbert’sNullstellensatz
Dilip P. Patil
§ 1Borsuk-UlamTheorem
§2 Borsuk’sNullstellensatzand itsEquivalents
§3 2-Fields
§4 DimensionandMultiplicity
§5 ProjectiveNullstellensatz
§6 Analogs ofHNS to 2fields
We end with the following two Remarks :
Remark 6.4The Theorems 6.1 and 6.2 und their proofs have analogues for arbitrary p-fields,where p is prime, 2 is replaced by p, see Remark 3.13.
Remark 6.5Analogous Theorems also holds for Zr-standard graded K-algebras can beproved. The corresponding geometric objects are projective algebraic sets inproducts Pn1(K)×·· ·×Pnr (K) of projective spaces. For such a algebra, insteadof a multiplicity, there are r multiplicities. Also one can forgo the standardgradation and the algebra generating elements may have of degree > 1. Themultiplicities are not nessarily integers, but may be rational numbers.
Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz