Invariant theory nqrsFrom Wikipedia, the free encyclopedia

Contents

1 Newtons identities 11.1 Mathematical statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Formulation in terms of symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Application to the roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Application to the characteristic polynomial of a matrix . . . . . . . . . . . . . . . . . . . 31.1.4 Relation with Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 A variant using complete homogeneous symmetric polynomials . . . . . . . . . . . . . . . 41.2.2 Expressing elementary symmetric polynomials in terms of power sums . . . . . . . . . . . 41.2.3 Expressing complete homogeneous symmetric polynomials in terms of power sums . . . . 41.2.4 Expressing power sums in terms of elementary symmetric polynomials . . . . . . . . . . . 51.2.5 Expressing power sums in terms of complete homogeneous symmetric polynomials . . . . 51.2.6 Expressions as determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Derivation of the identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 From the special case n = k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Comparing coecients in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 As a telescopic sum of symmetric function identities . . . . . . . . . . . . . . . . . . . . 9

1.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Nullform 112.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Osculant 123.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Perpetuant 134.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Polynomial ring 155.1 The polynomial ring K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.1.2 Degree of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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ii CONTENTS

5.1.3 Properties of K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Polynomial evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 The polynomial ring in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.2 The polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.3 Hilberts Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4 Properties of the ring extension R R[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5.1 Innitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5.2 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5.4 Noncommutative polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.5.5 Dierential and skew-polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Quantum invariant 246.1 List of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Quaternary cubic 277.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Sylvester pentahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Quippian 298.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Standard monomial theory 309.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

CONTENTS iii

9.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 1

Newtons identities

In mathematics,Newtons identities, also known as theNewtonGirard formulae, give relations between two typesof symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at theroots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P(counted with their multiplicity) in terms of the coecients of P, without actually nding those roots. These identitieswere found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They haveapplications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics,as well as further applications outside mathematics, including general relativity.

1.1 Mathematical statement

1.1.1 Formulation in terms of symmetric polynomialsLet x1, , xn be variables, denote for k 1 by pk(x1, , xn) the k-th power sum:

pk(x1; : : : ; xn) =Xn

i=1xki = x

k1 + + xkn;

and for k 0 denote by ek(x1, , xn) the elementary symmetric polynomial (that is, the sum of all distinct productsof k distinct variables), so

e0(x1; : : : ; xn) = 1;

e1(x1; : : : ; xn) = x1 + x2 + + xn;e2(x1; : : : ; xn) =

P1i n:

Then Newtons identities can be stated as

kek(x1; : : : ; xn) =kX

i=1

(1)i1eki(x1; : : : ; xn)pi(x1; : : : ; xn);

valid for all n 1 and k 1.Also, one has

0 =kX

i=kn(1)i1eki(x1; : : : ; xn)pi(x1; : : : ; xn);

1

2 CHAPTER 1. NEWTONS IDENTITIES

for all k > n 1.Concretely, one gets for the rst few values of k:

e1(x1; : : : ; xn) = p1(x1; : : : ; xn);

2e2(x1; : : : ; xn) = e1(x1; : : : ; xn)p1(x1; : : : ; xn) p2(x1; : : : ; xn);3e3(x1; : : : ; xn) = e2(x1; : : : ; xn)p1(x1; : : : ; xn) e1(x1; : : : ; xn)p2(x1; : : : ; xn) + p3(x1; : : : ; xn):

The form and validity of these equations do not depend on the number n of variables (although the point where theleft-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities inthe ring of symmetric functions. In that ring one has

e1 = p1;

2e2 = e1p1 p2;3e3 = e2p1 e1p2 + p3;4e4 = e3p1 e2p2 + e1p3 p4;

and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms ofthe pk; to be able to do the inverse, one may rewrite them as

p1 = e1;

p2 = e1p1 2e2;p3 = e1p2 e2p1 + 3e3;p4 = e1p3 e2p2 + e3p1 4e4;

...

In general, we have

pk(x1; : : : ; xn) = (1)k1kek(x1; : : : ; xn) +k1Xi=1

(1)k1+ieki(x1; : : : ; xn)pi(x1; : : : ; xn);

valid for all n 1 and k 1.Also, one has

pk(x1; : : : ; xn) =k1X

i=kn(1)k1+ieki(x1; : : : ; xn)pi(x1; : : : ; xn);

for all k > n 1.

1.1.2 Application to the roots of a polynomialThe polynomial with roots xi may be expanded as

nYi=1

(x xi) =nX

k=0

(1)n+kenkxk;

where the coecients ek(x1; : : : ; xn) are the symmetric polynomials dened above. Given the power sums of theroots

1.1. MATHEMATICAL STATEMENT 3

pk(x1; : : : ; xn) =nXi=1

xki ;

the coecients of the polynomial with roots x1; : : : ; xn may be expressed recursively in terms of the power sums as

e0 = 1;

e1 = p1;

e2 =1

2(e1p1 p2);

e3 =1

3(e2p1 e1p2 + p3);

e4 =1

4(e3p1 e2p2 + e1p3 p4);

...

Formulating polynomial this way is useful in using the method of Delves and Lyness[1] to nd the zeros of an analyticfunction.

1.1.3 Application to the characteristic polynomial of a matrixWhen the polynomial above is the characteristic polynomial of a matrix A (in particular when A is the companionmatrix of the polynomial), the roots xi are the eigenvalues of the matrix, counted with their algebraic multiplicity.For any positive integer k, the matrix Ak has as eigenvalues the powers xik, and each eigenvalue xi of A contributesits multiplicity to that of the eigenvalue xik of Ak. Then the coecients of the characteristic polynomial of Ak aregiven by the elementary symmetric polynomials in those powers xik. In particular, the sum of the xik, which is thek-th power sum sk of the roots of the characteristic polynomial of A, is given by its trace:

sk = tr(Ak) :

The Newton identities now relate the traces of the powers Ak to the coecients of the characteristic polynomial ofA. Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can beused to nd the characteristic polynomial by computing only the powers Ak and their traces.This computation requires computing the traces of matrix powers Ak and solving a triangular system of equations.Both can be done in complexity class NC (solving a triangular system can be done by divide-and-conquer). Therefore,characteristic polynomial of a matrix can be computed in NC. By the Cayley-Hamilton theorem, every matrix satisesits characteristic polynomial, and a simple transformation allows to nd the matrix inverse in NC.Rearranging the computations into an ecient form leads to the Fadeev-Leverrier algorithm (1840), a fast parallelimplementation of it is due to L. Csanky (1976). Its disadvantage is that it requires division by integers, so in generalthe eld should have characteristic, 0.

1.1.4 Relation with Galois theoryFor a given n, the elementary symmetric polynomials ek(x1,,xn) for k = 1,, n form an algebraic basis for the spaceof symmetric polynomials in x1,. xn: every polynomial expression in the xi that is invariant under all permutations ofthose variables is given by a polynomial expression in those elementary symmetric polynomials, and this expressionis unique up to equivalence of polynomial expressions. This is a general fact known as the fundamental theoremof symmetric polynomials, and Newtons identities provide explicit formulae in the case of power sum symmetricpolynomials. Applied to the monic polynomial tn +Pnk=1(1)kaktnk with all coecients ak considered as freeparameters, this means that every symmetric polynomial expression S(x1,,xn) in its roots can be expressed insteadas a polynomial expression P(a1,,an) in terms of its coecients only, in other words without requiring knowledgeof the roots. This fact also follows from general considerations in Galois theory (one views the ak as elements of a

4 CHAPTER 1. NEWTONS IDENTITIES

base eld with roots in an extension eld whose Galois group permutes them according to the full symmetric group,and the eld xed under all elements of the Galois group is the base eld).The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sumsymmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact therst n power sums also form an algebraic basis for the space of symmetric polynomials.

1.2 Related identitiesThere are a number of (families of) identities that, while they should be distinguished from Newtons identities, arevery closely related to them.

1.2.1 A variant using complete homogeneous symmetric polynomialsDenoting by hk the complete homogeneous symmetric polynomial that is the sum of all monomials of degree k,the power sum polynomials also satisfy identities similar to Newtons identities, but not involving any minus signs.Expressed as identities of in the ring of symmetric functions, they read

khk =kX

i=1

hkipi;

valid for all n k 1. Contrary to Newtons identities, the left-hand sides do not become zero for large k, and theright-hand sides contain ever more non-zero terms. For the rst few values of k, one has

h1 = p1;

2h2 = h1p1 + p2;

3h3 = h2p1 + h1p2 + p3:

These relations can be justied by an argument analogous to the one by comparing coecients in power series givenabove, based in this case on the generating function identity

1Xk=0

hk(X1; : : : ; Xn)tk =

nYi=1

1

1Xit :

Proofs of Newtons identities, like these given below, cannot be easily adapted to prove these variants of those iden-tities.

1.2.2 Expressing elementary symmetric polynomials in terms of power sumsAs mentioned, Newtons identities can be used to recursively express elementary symmetric polynomials in termsof power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring Q ofsymmetric functions with rational coecients:and so forth. Applied to a monic polynomial, these formulae express the coecients in terms of the power sums ofthe roots: replace each ei by ai and each pk by sk.

1.2.3 Expressing complete homogeneous symmetric polynomials in terms of power sumsThe analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, givingequationsand so forth, in which there are only plus signs. These expressions correspond exactly to the cycle index polynomialsof the symmetric groups, if one interprets the power sums pi as indeterminates: the coecient in the expression for hk

1.2. RELATED IDENTITIES 5

of any monomial p1m1p2m2plml is equal to the fraction of all permutations of k that have m1 xed points, m2 cyclesof length 2,, andml cycles of length l. Explicitly, this coecient can be written as 1/N whereN = li=1(mi! imi); thisN is the number permutations commuting with any given permutation of the given cycle type. The expressionsfor the elementary symmetric functions have coecients with the same absolute value, but a sign equal to the sign of, namely (1)m2+m4+....It can be proved by considering the following inductive step:

mf(m;m1; :::;mn) = f(m 1;m1 1; :::;mn) + :::+ f(m n;m1; :::;mn 1)

m1

nYi=1

1

imimi!+ :::+ nmn

nYi=1

1

imimi!= m

nYi=1

1

imimi!

1.2.4 Expressing power sums in terms of elementary symmetric polynomials

One may also use Newtons identities to express power sums in terms of symmetric polynomials, which does notintroduce denominators:

p1 = e1;

p2 = e21 2e2;

p3 = e31 3e2e1 + 3e3;

p4 = e41 4e2e21 + 4e3e1 + 2e22 4e4;

p5 = e51 5e2e31 + 5e3e21 + 5e22e1 5e4e1 5e3e2 + 5e5;

p6 = e61 6e2e41 + 6e3e31 + 9e22e21 6e4e21 12e3e2e1 + 6e5e1 2e32 + 3e23 + 6e4e2 6e6:

The rst four formulas were obtained by Albert Girard in 1629 (thus before Newton).[2]

The general formula (for all positive integers m and n) is:

pm =X

r1+2r2++nrn=mr10;:::;rn0

(1)mm(r1 + r2 + + rn 1)!r1!r2! rn!

nYi=1

(ei)ri

which can be proved by considering the following inductive step:

f(m; r1; ; rn) = f(m 1; r1 1; ; rn) + + f(m n; r1; ; rn 1)=

1

(r1 1)! rn! (m 1)(r1 + + rn 2)! + +1

r1! (rn 1)! (m n)(r1 + + rn 2)!

=1

r1! rn! [r1(m 1) + + rn(m n)] [r1 + + rn 2]!

=1

r1! rn! [m(r1 + + rn)m] [r1 + + rn 2]!

=m(r1 + + rn 1)!

r1! rn!

1.2.5 Expressing power sums in terms of complete homogeneous symmetric polynomials

Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to ex-press power sums in term of them:

6 CHAPTER 1. NEWTONS IDENTITIES

p1 = +h1;

p2 = h21 + 2h2;p3 = +h

31 3h2h1 + 3h3;

p4 = h41 + 4h2h21 4h3h1 2h22 + 4h4;p5 = +h

51 5h2h31 + 5h22h1 + 5h3h21 5h3h2 5h4h1 + 5h5;

p6 = h61 + 6h2h41 9h22h21 6h3h31 + 2h32 + 12h3h2h1 + 6h4h21 3h23 6h4h2 6h1h5 + 6h6;

and so on. Apart from the replacement of each ei by the corresponding hi, the only change with respect to the previousfamily of identities is in the signs of the terms, which in this case depend just on the number of factors present: thesign of the monomial li=1hmii is (1)m1+m2+m3+. In particular the above description of the absolute value of thecoecients applies here as well.The general formula (for all positive integers m and n) is:

pm = X

m1+2m2++nmn=mm10;:::;mn0

m(r1 + r2 + + rn 1)!r1!r2! rn!

nYi=1

(hi)ri

1.2.6 Expressions as determinantsOne can obtain explicit formulas for the above expressions in the form of determinants, by considering the rst n ofNewtons identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which theelementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramersrule to nd the solution for the nal unknown. For instance taking Newtons identities in the form

e1 = 1p1;

2e2 = e1p1 1p2;3e3 = e2p1 e1p2 + 1p3;

...nen = en1p1 en2p2 + + (1)ne1pn1 + (1)n1pnwe consider p1 , p2 , p3 , , (1)npn1 and pn as unknowns, and solve for the nal one, giving

pn =

1 0 e1e1 1 0 2e2e2 e1 1 3e3... . . . . . . ...

en1 e2 e1 nen

1 0 e1 1 0 e2 e1 1... . . . . . .

en1 e2 e1 (1)n1

1

=1

(1)n1

1 0 e1e1 1 0 2e2e2 e1 1 3e3... . . . . . . ...

en1 e2 e1 nen

=

e1 1 0 2e2 e1 1 0 3e3 e2 e1 1... . . . . . .

nen en1 e1

:

1.3. DERIVATION OF THE IDENTITIES 7

Solving for en instead of for pn is similar, as the analogous computations for the complete homogeneous symmetricpolynomials; in each case the details are slightly messier than the nal results, which are (Macdonald 1979, p. 20):

en =1

n!

p1 1 0 p2 p1 2 0 ... . . . . . .

pn1 pn2 p1 n 1pn pn1 p2 p1

pn = (1)n1

h1 1 0 2h2 h1 1 0 3h3 h2 h1 1... . . . . . .

nhn hn1 h1

hn =

1

n!

p1 1 0 p2 p1 2 0 ... . . . . . .

pn1 pn2 p1 1 npn pn1 p2 p1

:

Note that the use of determinants makes that the formula for hn has additional minus signs compared to the one foren , while the situation for the expanded form given earlier is opposite. As remarked in (Littlewood 1950, p. 84) onecan alternatively obtain the formula for hn by taking the permanent of the matrix for en instead of the determinant,and more generally an expression for any Schur polynomial can be obtained by taking the corresponding immanantof this matrix.

1.3 Derivation of the identitiesEach of Newtons identities can easily be checked by elementary algebra; however, their validity in general needs aproof. Here are some possible derivations.

1.3.1 From the special case n = kOne can obtain the k-th Newton identity in k variables by substitution into

kYi=1

(t xi) =kX

i=0

(1)kieki(x1; : : : ; xk)ti

as follows. Substituting xj for t gives

0 =kX

i=0

(1)kieki(x1; : : : ; xk)xji for1 j k

Summing over all j gives

0 = (1)kkek(x1; : : : ; xk) +kX

i=1

(1)kieki(x1; : : : ; xk)pi(x1; : : : ; xk);

where the terms for i = 0 were taken out of the sum because p0 is (usually) not dened. This equation immediatelygives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of

8 CHAPTER 1. NEWTONS IDENTITIES

degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities inn < k variables can be deduced by setting k n variables to zero. The k-th Newton identity in n > k variables containsmore terms on both sides of the equation than the one in k variables, but its validity will be assured if the coecientsof any monomial match. Because no individual monomial involves more than k of the variables, the monomial willsurvive the substitution of zero for some set of n k (other) variables, after which the equality of coecients is onethat arises in the k-th Newton identity in k (suitably chosen) variables.

1.3.2 Comparing coecients in seriesAnother derivation can be obtained by computations in the ring of formal power series R[[t]], where R is Z[x1,,xn], the ring of polynomials in n variables x1,, xn over the integers.Starting again from the basic relation

nYi=1

(t xi) =nX

k=0

(1)kaktnk

and reversing the polynomials by substituting 1/t for t and then multiplying both sides by tn to remove negativepowers of t, gives

nYi=1

(1 xit) =nX

k=0

(1)kaktk:

(the above computation should be performed in the eld of fractions of R[[t]]; alternatively, the identity can beobtained simply by evaluating the product on the left side)Swapping sides and expressing the ai as the elementary symmetric polynomials they stand for gives the identity

nXk=0

(1)kek(x1; : : : ; xn)tk =nYi=1

(1 xit):

One formally dierentiates both sides with respect to t, and then (for convenience) multiplies by t, to obtain

nXk=0

(1)kkek(x1; : : : ; xn)tk = tnXi=1

h(xi)

Yj 6=i(1 xjt)

i=

nXi=1

xit

1 xit

!Ynj=1

(1 xjt)

= 24 nXi=1

1Xj=1

(xit)j

35" nX`=0

(1)`e`(x1; : : : ; xn)t`#

=

24 1Xj=1

pj(x1; : : : ; xn)tj

35" nX`=0

(1)`1e`(x1; : : : ; xn)t`#;

where the polynomial on the right hand side was rst rewritten as a rational function in order to be able to factor outa product out of the summation, then the fraction in the summand was developed as a series in t, using the formula

X

1X = X +X2 +X3 +X4 +X5 +

and nally the coecient of each t j was collected, giving a power sum. (The series in t is a formal power series, butmay alternatively be thought of as a series expansion for t suciently close to 0, for those more comfortable with that;

1.4. SEE ALSO 9

in fact one is not interested in the function here, but only in the coecients of the series.) Comparing coecients oftk on both sides one obtains

(1)kkek(x1; : : : ; xn) =kX

j=1

(1)kj1pj(x1; : : : ; xn)ekj(x1; : : : ; xn);

which gives the k-th Newton identity.

1.3.3 As a telescopic sum of symmetric function identitiesThe following derivation, given essentially in (Mead, 1992), is formulated in the ring of symmetric functions forclarity (all identities are independent of the number of variables). Fix some k > 0, and dene the symmetric functionr(i) for 2 i k as the sum of all distinct monomials of degree k obtained by multiplying one variable raised tothe power i with k i distinct other variables (this is the monomial symmetric function m where is a hook shape(i,1,1,1)). In particular r(k) = pk; for r(1) the description would amount to that of ek, but this case was excludedsince here monomials no longer have any distinguished variable. All products pieki can be expressed in terms of ther(j) with the rst and last case being somewhat special. One has

pieki = r(i) + r(i+ 1) for1 < i < k

since each product of terms on the left involving distinct variables contributes to r(i), while those where the variablefrom pi already occurs among the variables of the term from eki contributes to r(i + 1), and all terms on the right areso obtained exactly once. For i = k one multiplies by e0 = 1, giving trivially

pke0 = pk = r(k)

Finally the product p1ek for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remainingcontributions produce k times each monomial of ek, since any one of the variables may come from the factor p1; thus

p1ek1 = kek + r(2)

The k-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of theform r(i) cancel out.

1.4 See also Power sum symmetric polynomial Elementary symmetric polynomial Symmetric function Fluid solutions, an article giving an application of Newtons identities to computing the characteristic polyno-mial of the Einstein tensor in the case of a perfect uid, and similar articles on other types of exact solutionsin general relativity.

1.5 References[1] Delves, L. M. (1967). A Numerical Method of Locating the Zeros of an Analytic Function. Mathematics of Computation

21: 543560. doi:10.2307/2004999.

[2] Tignol, Jean-Pierre (2004). Galois theory of algebraic equations (Reprinted. ed.). River Edge, NJ: World Scientic. pp.3738. ISBN 981-02-4541-6.

10 CHAPTER 1. NEWTONS IDENTITIES

Tignol, Jean-Pierre (2001). Galois theory of algebraic equations. Singapore: World Scientic. ISBN 978-981-02-4541-2.

Bergeron, F.; Labelle, G.; and Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge:Cambridge University Press. ISBN 978-0-521-57323-8.

Cameron, Peter J. (1999). Permutation Groups. Cambridge: Cambridge University Press. ISBN 978-0-521-65378-7.

Cox, David; Little, John, and O'Shea, Donal (1992). Ideals, Varieties, and Algorithms. New York: Springer-Verlag. ISBN 978-0-387-97847-5.

Eppstein, D.; Goodrich, M. T. (2007). Space-ecient straggler identication in round-trip data streams viaNewtons identities and invertible Bloom lters. Algorithms and Data Structures, 10th International Workshop,WADS 2007. Springer-Verlag, Lecture Notes in Computer Science 4619. pp. 637648. arXiv:0704.3313

Littlewood, D. E. (1950). The theory of group characters and matrix representations of groups. Oxford: OxfordUniversity Press. viii+310. ISBN 0-8218-4067-3.

Macdonald, I. G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs.Oxford: The Clarendon Press, Oxford University Press. viii+180. ISBN 0-19-853530-9. MR 84g:05003.

Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Sec-ond ed.). New York: Oxford Science Publications. The Clarendon Press, Oxford University Press. p. x+475.ISBN 0-19-853489-2. MR 96h:05207.

Mead, D.G. (1992-10). Newtons Identities. The American Mathematical Monthly (Mathematical Associ-ation of America) 99 (8): 749751. doi:10.2307/2324242. JSTOR 2324242. Check date values in: |date=(help)

Stanley, Richard P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press. ISBN 0-521-56069-1. (hardback). ISBN 0-521-78987-7 (paperback).

Sturmfels, Bernd (1992). Algorithms in Invariant Theory. New York: Springer-Verlag. ISBN 978-0-387-82445-1.

Tucker, Alan (1980). Applied Combinatorics (5/e ed.). New York: Wiley. ISBN 978-0-471-73507-6.

1.6 External links NewtonGirard formulas on MathWorld A Matrix Proof of Newtons Identities in Mathematics Magazine Application on the number of real roots

Chapter 2

Nullform

In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of thegroup vanish. Nullforms were introduced by Hilbert (1893). (Dieudonn & Carrell 1970, 1971, p.57).

2.1 References Dieudonn, Jean A.; Carrell, James B. (1970), Invariant theory, old and new, Advances in Mathematics 4:180, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

Dieudonn, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

Hilbert, David (1893), Ueber die vollen Invariantensysteme,Mathematische Annalen (Springer Berlin / Hei-delberg) 42: 313373, doi:10.1007/BF01444162, ISSN 0025-5831

11

Chapter 3

Osculant

In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurfacethat vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning thatthey have a common point where they meet to unusually high order.

3.1 References Salmon, George (1885) [1859], Lessons introductory to the modern higher algebra (4th ed.), Dublin, Hodges,Figgis, and Co., ISBN 978-0-8284-0150-0

12

Chapter 4

Perpetuant

In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or innite degree.More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary formstabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space arecalled perpetuants. Perpetuants were introduced and named by Sylvester (1882, p.105). MacMahon (1884, 1885,1894) and Stroh (1888) classied the perpetuants. Elliott (1907) describes the early history of perpetuants and givesan annotated bibliography. There are very few papers after about 1910 discussing perpetuants; (Littlewood 1944) isone of the few exceptions.MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weightw is the coecient of xw of

x2n11

(1 x2)(1 x3) (1 xn)

For n=1 there is just one perpetuant, of weight 0, and for n=2 the number is given by the coecient of xw of x2/(1-x2).

4.1 References Cayley, Arthur (1884), A Memoir on Seminvariants, American Journal of Mathematics (The Johns HopkinsUniversity Press) 7 (1): 125, doi:10.2307/2369456, ISSN 0002-9327

Elliott, Edwin Bailey (1895), An introduction to the algebra of quantics, Oxford, Clarendon Press, Reprintedby Chelsea Scientic Books 1964

Elliott, Edwin Bailey (1907), On Perpetuants and Contra-Perpetuants, Proc. London Math. Soc. 4 (1):228246, doi:10.1112/plms/s2-4.1.228

Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge University Press

Littlewood, D. E. (1944), Invariant theory, tensors and group characters, Philosophical Transactions of theRoyal Society of London. Series A.Mathematical and Physical Sciences 239: 305365, doi:10.1098/rsta.1944.0001,ISSN 0080-4614, MR 0010594

MacMahon, P. A. (1884), On Perpetuants, American Journal of Mathematics (The Johns Hopkins UniversityPress) 7 (1): 2646, doi:10.2307/2369457, ISSN 0002-9327

MacMahon, P. A. (1885), A Second Paper on Perpetuants, American Journal of Mathematics (The JohnsHopkins University Press) 7 (3): 259263, doi:10.2307/2369271, ISSN 0002-9327

MacMahon, P. A. (1894), The Perpetuant Invariants of Binary Quantics, Proc. London Math. Soc. 26 (1):262284, doi:10.1112/plms/s1-26.1.262

13

14 CHAPTER 4. PERPETUANT

Stroh, E. (1888), Ueber eine fundamentale Eigenschaft des Ueberschiebungs-processes und deren Verw-erthung in der Theorie der binren Formen, Mathematische Annalen (Springer Berlin / Heidelberg) 33: 61107, doi:10.1007/bf01444111, ISSN 0025-5831

Sylvester, James Joseph (1882), On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Or-der,American Journal ofMathematics (The JohnsHopkinsUniversity Press) 5 (1): 79136, doi:10.2307/2369536,ISSN 0002-9327

Chapter 5

Polynomial ring

In mathematics, especially in the eld of abstract algebra, a polynomial ring is a ring formed from the set ofpolynomials in one or more indeterminates (traditionally also called variables) with coecients in another ring, oftena eld. Polynomial rings have inuenced much of mathematics, from the Hilbert basis theorem, to the constructionof splitting elds, and to the understanding of a linear operator. Many important conjectures involving polynomialrings, such as Serres problem, have inuenced the study of other rings, and have inuenced even the denition ofother rings, such as group rings and rings of formal power series.A closely related notion is that of the ring of polynomial functions on a vector space.

5.1 The polynomial ring K[X]

5.1.1 Denition

The polynomial ring, K[X], in X over a eld K is dened[1] as the set of expressions, called polynomials in X, ofthe form

p = p0 + p1X + p2X2 + + pm1Xm1 + pmXm;

where p0, p1,, p, the coecients of p, are elements of K, and X, X 2, are formal symbols (the powers of X"). Byconvention, X 0 = 1, X 1 = X, and the product of the powers of X is dened by the familiar formula

XkX l = Xk+l;

where k and l are any natural numbers. The symbol X is called an indeterminate[2] or variable.[3]

Two polynomials are dened to be equal if and only if the corresponding coecients for each power of X are equal,however terms with zero coecient, 0X k, may be added or omitted.This terminology is suggested by real or complex polynomial functions. However, in general, X and its powers, X k,are treated as formal symbols, not as elements of the eld K or functions over it. One can think of the ring K[X] asarising from K by adding one new element X that is external to K and requiring that X commute with all elements ofK.Polynomials in X are added and multiplied according to the ordinary rules for manipulating algebraic expressions,creating the structure of a ring. Specically, if

p = p0 + p1X + p2X2 + + pmXm;

and

15

16 CHAPTER 5. POLYNOMIAL RING

q = q0 + q1X + q2X2 + + qnXn;

then

p+ q = r0 + r1X + r2X2 + + rkXk;

and

pq = s0 + s1X + s2X2 + + slX l;

where

ri = pi + qi

and

si = p0qi + p1qi1 + + piq0:If necessary, the polynomials p and q are extended by adding dummy terms with zero coecients, so that theexpressions for ri and si are always dened.A more rigorous, but less intuitive treatment[4] denes a polynomial as an innite tuple, or ordered sequence of ele-ments of K, (p0, p1, p2, ) having the property that only a nite number of the elements are nonzero, or equivalently,a sequence for which there is some m so that pn = 0 for n>m. In this case, the expression

p0 + p1X + p2X2 + + pmXm

is considered an alternate notation for the sequence (p0, p1, p2, pm, 0, 0, ).More generally, the eld K can be replaced by any commutative ring R when taking the same construction as above,giving rise to the polynomial ring over R, which is denoted R[X].

5.1.2 Degree of a polynomialThe degree of a polynomial p, written deg(p) is the largest k such that the coecient of X k is not zero.[5] In this casethe coecient pk is called the leading coecient.[6] In the special case of zero polynomial, all of whose coecientsare zero, the degree has been variously left undened,[7] dened to be 1,[8] or dened to be a special symbol .[9]

If K is a eld, or more generally an integral domain, then from the denition of multiplication,[10]

deg(pq) = deg(p) + deg(q):

It follows immediately that if K is an integral domain then so is K[X].[11]

5.1.3 Properties of K[X]Factorization in K[X]

The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can beuniquely factored into a product of primes this statement is now called the fundamental theorem of arithmetic.The proof is based on Euclids algorithm for nding the greatest common divisor of natural numbers. At each step

5.1. THE POLYNOMIAL RING K[X] 17

of this algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainderfrom the division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division withthe remainder can also be dened for polynomials: given two polynomials p and q, where q 0, one can write

p = uq + r;

where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and a de-composition with these properties is unique. The quotient and the remainder are found using the polynomial longdivision. The degree of the polynomial now plays a role similar to the absolute value of an integer: it is strictly lessin the remainder r than it is in q, and when repeating this step such decrease cannot go on indenitely. Thereforeeventually some division will be exact, at which point the last non-zero remainder is the greatest common divisorof the initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneouslyrigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In factthere exist other commutative rings than Z and K[X] that similarly admit an analogue of the Euclidean algorithm;all such rings are called Euclidean rings. Rings for which there exists unique (in an appropriate sense) factoriza-tion of nonzero elements into irreducible factors are called unique factorization domains or factorial rings; the givenconstruction shows that all Euclidean rings, and in particular Z and K[X], are unique factorization domains.Another corollary of the polynomial division with the remainder is the fact that every proper ideal I of K[X] isprincipal, i.e. I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a principalideal domain, and for the same reason every Euclidean domain is a principal ideal domain. Also every principalideal domain is a unique-factorization domain. These deductions make essential use of the fact that the polynomialcoecients lie in a eld, namely in the polynomial division step, which requires the leading coecient of q, whichis only known to be non-zero, to have an inverse. If R is an integral domain that is not a eld then R[X] is neither aEuclidean domain nor a principal ideal domain; however it could still be a unique factorization domain (and will beso if and only it R itself is a unique factorization domain, for instance if it is Z or another polynomial ring).

Quotient ring of K[X]

The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that any commu-tative ring L containing K and generated as a ring by a single element in addition to K can be described using K[X].In particular, this applies to nite eld extensions of K.Suppose that a commutative ring L contains K and there exists an element of L such that the ring L is generated by over K. Thus any element of L is a linear combination of powers of with coecients in K. Then there is a uniquering homomorphism from K[X] into L which does not aect the elements of K itself (it is the identity map on K)and maps each power of X to the same power of . Its eect on the general polynomial amounts to replacing X with":

'(amXm + am1Xm1 + + a1X + a0) = amm + am1m1 + + a1 + a0:

By the assumption, any element of L appears as the right hand side of the last expression for suitable m and elementsa0, , am of K. Therefore, is surjective and L is a homomorphic image of K[X]. More formally, let Ker be thekernel of . It is an ideal of K[X] and by the rst isomorphism theorem for rings, L is isomorphic to the quotientof the polynomial ring K[X] by the ideal Ker . Since the polynomial ring is a principal ideal domain, this ideal isprincipal: there exists a polynomial pK[X] such that

L ' K[X]/(p):

A particularly important application is to the case when the larger ring L is a eld. Then the polynomial p must beirreducible. Conversely, the primitive element theorem states that any nite separable eld extension L/K can begenerated by a single element L and the preceding theory then gives a concrete description of the eld L as thequotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration,the eld C of complex numbers is an extension of the eld R of real numbers generated by a single element i suchthat i2 + 1 = 0. Accordingly, the polynomial X2 + 1 is irreducible over R and

18 CHAPTER 5. POLYNOMIAL RING

C ' R[X]/(X2 + 1):More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commuteswith all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:

: K[X] ! A; (X) = a:This homomorphism is given by the same formula as before, but it is not surjective in general. The existence anduniqueness of such a homomorphism expresses a certain universal property of the ring of polynomials in one variableand explains ubiquity of polynomial rings in various questions and constructions of ring theory and commutativealgebra.

5.1.4 ModulesThe structure theorem for nitely generated modules over a principal ideal domain applies over K[X]. This meansthat every nitely generated module over K[X] may be decomposed into a direct sum of a free module and nitelymany modules of the formK[X]/hP ki , where P is an irreducible polynomial over K and k a positive integer.

5.2 Polynomial evaluationLet K be a eld or, more generally, a commutative ring, and R a ring containing K. For any polynomial P in K[X]and any element a in R, the substitution of X by a in P denes an element of R, which is denoted P(a). This elementis obtained by, after the substitution, carrying on, in R, the operations indicated by the expression of the polynomial.This computation is called the evaluation of P at a. For example, if we have

P = X2 1;we have

P (3) = 32 1 = 8;P (X2 + 1) = (X2 + 1)2 1 = X4 + 2X2(in the rst example R = K, and in the second one R = K[X]). Substituting X by itself results in

P = P (X);

explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent.For every a in R, the map P 7! P (a) denes a ring homomorphism from K[X] into R.The polynomial function dened by a polynomial P is the function from K into K that is dened by x 7! P (x): If Kis an innite eld, two dierent polynomials dene dierent polynomial functions, but this property is false for niteelds. For example, if K is a eld with q elements, then the polynomials 0 and Xq-X both dene the zero function.

5.3 The polynomial ring in several variables

5.3.1 PolynomialsA polynomial in n variables X1, , Xn with coecients in a eld K is dened analogously to a polynomial in onevariable, but the notation is more cumbersome. For any multi-index = (1, , n), where each i is a non-negativeinteger, let

5.3. THE POLYNOMIAL RING IN SEVERAL VARIABLES 19

X =

nYi=1

Xii = X11 : : : X

nn :

The product X is called themonomial of multidegree . A polynomial is a nite linear combination of monomialswith coecients in K

p =X

pX;

where p = p1;:::;n 2 K; and only nitely many coecients p are dierent from 0. The degree of a monomialX, frequently denoted ||, is dened as

jj =nXi=1

i;

and the degree of a polynomial p is the largest degree of a monomial occurring with non-zero coecient in theexpansion of p.

5.3.2 The polynomial ring

Polynomials in n variables with coecients in K form a commutative ring denoted K[X1,, Xn], or sometimesK[X], where X is a symbol representing the full set of variables, X = (X1,, Xn), and called the polynomial ring inn variables. The polynomial ring in n variables can be obtained by repeated application of K[X] (the order by whichis irrelevant). For example, K[X1, X2] is isomorphic to K[X1][X2]. This ring plays fundamental role in algebraicgeometry. Many results in commutative and homological algebra originated in the study of its ideals and modulesover this ring.A polynomial ring with coecients in Z is the free commutative ring over its set of variables.

5.3.3 Hilberts Nullstellensatz

Main article: Hilberts Nullstellensatz

A group of fundamental results concerning the relation between ideals of the polynomial ring K[X1,, Xn] andalgebraic subsets of Kn originating with David Hilbert is known under the name Nullstellensatz (literally: zero-locus theorem).

(Weak form, algebraically closed eld of coecients). Let K be an algebraically closed eld. Then everymaximal ideal m of K[X1,, Xn] has the form

m = (X1 a1; : : : ; Xn an); a = (a1; : : : ; an) 2 Kn:

(Weak form, any eld of coecients). Let k be a eld, K be an algebraically closed eld extension of k, and Ibe an ideal in the polynomial ring k[X1,, Xn]. Then I contains 1 if and only if the polynomials in I do nothave any common zero in Kn.

(Strong form). Let k be a eld, K be an algebraically closed eld extension of k, I be an ideal in the polynomialring k[X1,, Xn],and V(I) be the algebraic subset of Kn dened by I. Suppose that f is a polynomial whichvanishes at all points of V(I). Then some power of f belongs to the ideal I:

20 CHAPTER 5. POLYNOMIAL RING

fm 2 I; some form 2 N:

Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As acorollary of this form of Nullstellensatz, there is a bijective correspondence between the radical idealsof K[X1,, Xn] for an algebraically closed eld K and the algebraic subsets of the n-dimensional anespace Kn. It arises from the map

I 7! V (I); I K[X1; : : : ; Xn]; V (I) Kn:

The prime ideals of the polynomial ring correspond to irreducible subvarieties of Kn.

5.4 Properties of the ring extension R R[X]One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings. Thenotation R S indicates that a ring R is a subring of a ring S. In this case S is called an overring of R and one speaksof a ring extension. This works particularly well for polynomial rings and allows one to establish many importantproperties of the ring of polynomials in several variables over a eld, K[X1,, Xn], by induction in n.

5.4.1 Summary of the resultsIn the following properties, R is a commutative ring and S = R[X1,, Xn] is the ring of polynomials in n variablesover R. The ring extension R S can be built from R in n steps, by successively adjoining X1,, Xn. Thus to establisheach of the properties below, it is sucient to consider the case n = 1.

If R is an integral domain then the same holds for S.

If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma.

Hilberts basis theorem: If R is a Noetherian ring, then the same holds for S.

Suppose that R is a Noetherian ring of nite global dimension. Then

gl dimR[X1; : : : ; Xn] = gl dimR+ n:

An analogous result holds for Krull dimension.

5.5 GeneralizationsPolynomial rings have been generalized in a great many ways, including polynomial rings with generalized exponents,power series rings, noncommutative polynomial rings, and skew-polynomial rings.

5.5.1 Innitely many variablesThe possibility to allow an innite set of indeterminates is not really a generalization, as the ordinary notion of poly-nomial ring allows for it. It is then still true that each monomial involves only a nite number of indeterminates(so that its degree remains nite), and that each polynomial is a linear combination of monomials, which by deni-tion involves only nitely many of them. This explains why such polynomial rings are relatively seldom considered:

5.5. GENERALIZATIONS 21

each individual polynomial involves only nitely many indeterminates, and even any nite computation involvingpolynomials remains inside some subring of polynomials in nitely many indeterminates.In the case of innitely many indeterminates, one can consider a ring strictly larger than the polynomial ring butsmaller than the power series ring, by taking the subring of the latter formed by power series whose monomials havea bounded degree. Its elements still have a nite degree and are therefore are somewhat like polynomials, but it ispossible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a rolein constructing the ring of symmetric functions.

5.5.2 Generalized exponentsMain article: Monoid ring

A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas foraddition and multiplication make sense as long as one can add exponents: XiXj = Xi+j . A set for which additionmakes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R whichare nonzero at only nitely many places can be given the structure of a ring known as R[N], the monoid ring of Nwith coecients in R. The addition is dened component-wise, so that if c = a+b, then cn = an + bn for every n in N.The multiplication is dened as the Cauchy product, so that if c = ab, then for each n in N, cn is the sum of all aibjwhere i, j range over all pairs of elements of N which sum to n.When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:

Xn2N

anXn

and then the formulas for addition and multiplication are the familiar:

Xn2N

anXn

!+

Xn2N

bnXn

!=Xn2N

(an + bn)Xn

and

Xn2N

anXn

! Xn2N

bnXn

!=Xn2N

0@ Xi+j=n

aibj

1AXnwhere the latter sum is taken over all i, j in N that sum to n.Some authors such as (Lang 2002, II,3) go so far as to take this monoid denition as the starting point, and regularsingle variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials inseveral variables simply take N to be the direct product of several copies of the monoid of non-negative integers.Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negativerational numbers, (Osbourne 2000, 4.4).

5.5.3 Power seriesMain article: Formal power series

Power series generalize the choice of exponent in a dierent direction by allowing innitely many nonzero terms. Thisrequires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy productare nite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent innite sums.For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series isdened as the set of functions from N to a ring R with addition component-wise, and multiplication given by theCauchy product. The ring of power series can be seen as the completion of the polynomial ring.

22 CHAPTER 5. POLYNOMIAL RING

5.5.4 Noncommutative polynomial ringsMain article: Free algebra

For polynomial rings of more than one variable, the products XY and YX are simply dened to be equal. A moregeneral notion of polynomial ring is obtained when the distinction between these two formal products is maintained.Formally, the polynomial ring in n noncommuting variables with coecients in the ring R is the monoid ring R[N],where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of nsymbols, with multiplication given by concatenation. Neither the coecients nor the variables need commute amongstthemselves, but the coecients and variables commute with each other.Just as the polynomial ring in n variables with coecients in the commutative ring R is the free commutative R-algebraof rank n, the noncommutative polynomial ring in n variables with coecients in the commutative ring R is the freeassociative, unital R-algebra on n generators, which is noncommutative when n > 1.

5.5.5 Dierential and skew-polynomial ringsMain article: Ore extension

Other generalizations of polynomials are dierential and skew-polynomial rings.A dierential polynomial ring is a ring of dierential operators formed from a ring R and a derivation of R into R.This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate onR by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation(ab) = a(b) + (a)b may be rewritten

X a = a X + (a):

This relation may be extended to dene a skew multiplication between two polynomials in X with coecients in R,which make them a non-commutative ring.The standard example, called a Weyl algebra, takes R to be a (usual) polynomial ring k[Y], and to be the standardpolynomial derivative @@Y . Taking a =Y in the above relation, one gets the canonical commutation relation,XY Y X= 1. Extending this relation by associativity and distributivity allows to construct explicitly the Weyl algebra.(Lam2001, 1,ex1.9).The skew-polynomial ring is dened similarly for a ring R and a ring endomorphism f of R, by extending themultiplication from the relationXr = f(r)X to produce an associative multiplication that distributes over the standardaddition. More generally, given a homomorphismF from themonoidN of the positive integers into the endomorphismring of R, the formula Xnr = F(n)(r)Xn allows to construct a skew-polynomial ring.(Lam 2001, 1,ex 1.11) Skewpolynomial rings are closely related to crossed product algebras.

5.6 See also Additive polynomial Laurent polynomial

5.7 References[1] Following Herstein p. 153

[2] Herstein, Hall p. 73

[3] Lang p. 97

[4] Following Hall p.72-73

5.7. REFERENCES 23

[5] Herstein p. 154

[6] Lang p.100

[7] Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), Calculus Single Variable, John Wiley & Sons, p. 31, ISBN9780470647707.

[8] Sendra, J. Rafael; Winkler, Franz; Prez-Diaz, Sonia (2007), Rational Algebraic Curves: A Computer Algebra Approach,Algorithms and Computation in Mathematics 22, Springer, p. 250, ISBN 9783540737247.

[9] Eves, Howard Whitley (1980), Elementary Matrix Theory, Dover, p. 183, ISBN 9780486150277.

[10] Herstein p.155, 162

[11] Herstein p.162

Hall, F. M. (1969). Section 3.6. An Introduction to Abstract Algebra 2. Cambridge University Press. ISBN0521084849.

Herstein, I. N. (1975). Section 3.9. Topics in Algebra. Wiley. ISBN 0471010901. Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN978-0-387-95325-0

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics 196, Berlin, New York:Springer-Verlag, ISBN 978-0-387-98934-1, MR 1757274

Chapter 6

Quantum invariant

In the mathematical eld of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jonespolynomial of surgery presentations of the knot complement.[1] [2] [3]

6.1 List of invariants Finite type invariant Kontsevich invariant Kashaevs invariant WittenReshetikhinTuraev invariant (ChernSimons) Invariant dierential operator[4]

RozanskyWitten invariant Vassiliev knot invariant Dehn invariant LMO invariant [5]

TuraevViro invariant DijkgraafWitten invariant [6]

ReshetikhinTuraev invariant Tau-invariant I-Invariant Klein J-invariant Quantum isotopy invariant [7]

ErmakovLewis invariant Hermitian invariant GoussarovHabiro theory of nite-type invariant Linear quantum invariant (orthogonal function invariant) MurakamiOhtsuki TQFT Generalized Casson invariant

24

6.2. SEE ALSO 25

Casson-Walker invariant KhovanovRozansky invariant HOMFLY polynomial K-theory invariants AtiyahPatodiSinger eta invariant Link invariant [8]

Casson invariant SeibergWitten invariant GromovWitten invariant Arf invariant Hopf invariant

6.2 See also Invariant theory Framed knot ChernSimons theory Algebraic geometry Seifert surface Geometric invariant theory

6.3 References[1] Reshetikhin, N. & Turaev, V. (1991). Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.

103 (1): 547. doi:10.1007/BF01239527. Retrieved 4 December 2012.

[2] Kontsevich, Maxim (1993). Vassilievs knot invariants. Adv. Soviet Math. 16: 137.

[3] Watanabe, Tadayuki (2007). Knotted trivalent graphs and construction of the LMO invariant from triangulations. OsakaJ. Math. 44 (2): 351. Retrieved 4 December 2012.

[4] [math/0406194] Invariant dierential operators for quantum symmetric spaces, II

[5] [math/0009222v1] Topological quantum eld theory and hyperk\"ahler geometry

[6] http://hal.archives-ouvertes.fr/docs/00/09/02/99/PDF/equality_arxiv_1.pdf

[7] http://knot.kaist.ac.kr/7thkgtf/Lawton1.pdf

[8] Invariants of 3-manifolds via link polynomials and quantum groups - Springer

6.4 Further reading Freedman, Michael H. (1990). Topology of 4-manifolds. Princeton, N.J: Princeton University Press. ISBN0691085773.

Ohtsuki, Tomotada (December 2001). Quantum Invariants. World Scientic Publishing Company. ISBN9789810246754.

26 CHAPTER 6. QUANTUM INVARIANT

6.5 External links Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev

Chapter 7

Quaternary cubic

In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros forma cubic surface in 3-dimensional projective space.

7.1 InvariantsSalmon (1860) and Clebsch (1861, 1861b) studied the ring of invariants of a quaternary cubic, which is a ringgenerated by invariants of degrees 8, 16, 24, 32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generatea polynomial ring. The generator of degree 100 is a skew invariant, whose square is a polynomial in the othergenerators given explicitly by Salmon. Salmon also gave an explicit formula for the discriminant as a polynomial inthe generators, though Edge (1980) pointed out that the formula has a widely-copied misprint in it.

7.2 Sylvester pentahedronA generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cuberoots of unity. This was conjectured by Sylvester in 1851, and proven 10 years later by Clebsch. The union of the 5planes where these 5 linear forms vanish is called the Sylvester pentahedron.

7.3 See also Ternary cubic

Ternary quartic

Invariants of a binary form

7.4 References Clebsch, A. (1861), Zur Theorie der algebraischer Flchen, Journal fr die reine und angewandteMathematik58: 93108, ISSN 0075-4102

Clebsch, A. (1861), Ueber eine Transformation der homogenen Funktionen dritter Ordnung mit vier Vern-derlichen, Journal fr die reine und angewandte Mathematik 58: 109126, doi:10.1515/crll.1861.58.109,ISSN 0075-4102

Edge, W. L. (1980), The Discriminant of a Cubic Surface, Proceedings of the Royal Irish Academy (RoyalIrish Academy) 80A (1): 7578, ISSN 0035-8975

27

28 CHAPTER 7. QUATERNARY CUBIC

Salmon, George (1860), On Quaternary Cubics, Philosophical Transactions of the Royal Society (The RoyalSociety) 150: 229239, doi:10.1098/rstl.1860.0015, ISSN 0080-4614

Schmitt, Alexander (1997), Quaternary cubic forms and projective algebraic threefolds, L'EnseignementMathmatique. Revue Internationale. IIe Srie 43 (3): 253270, ISSN 0013-8584, MR 1489885

Chapter 8

Quippian

In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Cayley (1857) anddiscussed by Dolgachev (2012, p.157). In the same paper Cayley also introduced another similar invariant that hecalled the pippian, now called the Cayleyan.

8.1 See also Glossary of classical algebraic geometry

8.2 References Cayley, Arthur (1857), A Memoir on Curves of the Third Order, Philosophical Transactions of the Royal

Society of London (The Royal Society) 147: 415446, doi:10.1098/rstl.1857.0021, ISSN 0080-4614

Dolgachev, Igor V. (2012), Classical Algebraic Geometry: a modern view, Cambridge University Press, ISBN978-1-107-01765-8

29

Chapter 9

Standard monomial theory

In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized agvariety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standardmonomials. Many of the results have been extended to KacMoody algebras and their groups.There are monographs on standard monomial theory by Lakshmibai & Raghavan (2008) and Seshadri (2007) andsurvey articles by V. Lakshmibai, C. Musili, and C. S. Seshadri (1979) and V. Lakshmibai and C. S. Seshadri (1991)One of important open problems is to give a completely geometric construction of the theory.[1]

9.1 HistoryAlfred Young (1928) introduced monomials associated to standard Young tableaux. Hodge (1943) (see also (Hodge& Pedoe 1994, p.378)) used Youngs monomials, which he called standard power products, named after standardtableaux, to give a basis for the homogeneous coordinate rings of complex Grassmannians. Seshadri (1978) initiated aprogram, called standard monomial theory, to extend Hodges work to varieties G/P, for P any parabolic subgroupof any reductive algebraic group in any characteristic, by giving explicit bases using standard monomials for sectionsof line bundles over these varieties. The case of Grassmannians studied by Hodge corresponds to the case whenG is a special linear group in characteristic 0 and P is a maximal parabolic subgroup. Seshadri was soon joined inthis eort by V. Lakshmibai and Chitikila Musili. They worked out standard monomial theory rst for minusculerepresentations of G and then for groups G of classical type, and formulated several conjectures describing it formore general cases. Littelmann (1998) proved their conjectures using the Littelmann path model, in particular givinga uniform description of standard monomials for all reductive groups.Lakshmibai (2003) and Musili (2003) and Seshadri (2012) give detailed descriptions of the early development ofstandard monomial theory.

9.2 Applications Since the sections of line bundles over generalized ag varieties tend to form irreducible representations of thecorresponding algebraic groups, having an explicit basis of standard monomials allows one to give characterformulas for these representations. Similarly one gets character formulas for Demazure modules. The explicitbases given by standard monomial theory are closely related to crystal bases and Littelmann path models ofrepresentations.

Standard monomial theory allows one to describe the singularities of Schubert varieties, and in particularsometimes proves that Schubert varieties are normal or CohenMacaulay. .

Standard monomial theory can be used to prove Demazures conjecture.

Standard monomial theory proves the Kempf vanishing theorem and other vanishing theorems for the highercohomology of eective line bundles over Schubert varieties.

30

9.3. NOTES 31

Standard monomial theory gives explicit bases for some rings of invariants in invariant theory.

Standard monomial theory gives generalizations of the LittlewoodRichardson rule about decompositions oftensor products of representations to all reductive algebraic groups.

Standard monomial theory can be used to prove the existence of good ltrations on some representations ofreductive algebraic groups in positive characteristic.

9.3 Notes[1] M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651680.

9.4 References Hodge, W. V. D. (1943), Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. 39(1): 2230, doi:10.1017/S0305004100017631, MR 0007739

Hodge, W. V. D.; Pedoe, Daniel (1994) [1952], Methods of Algebraic Geometry: Volume 2 Book III: Gen-eral theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties., CambridgeMathematical Library, Cambridge University Press, ISBN 978-0-521-46901-2, MR 0048065

Lakshmibai, V.; Musili, C.; Seshadri, C. S. (1979), Geometry of G/P, American Mathematical Society.Bulletin. New Series 1 (2): 432435, doi:10.1090/S0273-0979-1979-14631-7, ISSN 0002-9904, MR 520081

Lakshmibai, Venkatramani; Raghavan, KomaranapuramN. (2008), Standard monomial theory, Encyclopaediaof Mathematical Sciences 137, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-76757-2, ISBN978-3-540-76756-5, MR 2388163

Lakshmibai, V.; Seshadri, C. S. (1991), Standard monomial theory, in Ramanan, S.; Musili, C.; Kumar, N.Mohan, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Madras: ManojPrakashan, pp. 279322, MR 1131317

Lakshmibai, V. (2003), The development of standard monomial theory. II, A tribute to C. S. Seshadri (Chen-nai, 2002), Trends Math., Basel, Boston, Berlin: Birkhuser, pp. 283309, ISBN 978-3-7643-0444-7, MR2017589

Littelmann, Peter (1998), The path model, the quantum Frobenius map and standard monomial theory, inCarter, Roger W.; Saxl, Jan, Algebraic groups and their representations (Cambridge, 1997), NATO AdvancedScience Institutes Series C: Mathematical and Physical Sciences 517, Dordrecht: Kluwer Academic PublishersGroup, pp. 175212, MR 1670770

Littelmann, Peter (1998), Contracting modules and standard monomial theory for symmetrizable Kac-Moodyalgebras, Journal of the AmericanMathematical Society 11 (3): 551567, doi:10.1090/S0894-0347-98-00268-9, ISSN 0894-0347, MR 1603862

Musili, C. (2003), The development of standard monomial theory. I, A tribute to C. S. Seshadri (Chen-nai, 2002), Trends Math., Basel, Boston, Berlin: Birkhuser, pp. 385420, ISBN 978-3-7643-0444-7, MR2017594

Seshadri, C. S. (1978), Geometry of G/P. I. Theory of standard monomials for minuscule representations,in Ramanathan, K. G., C. P. Ramanujam---a tribute, Tata Institute of Fundamental Research Studies in Math-ematics 8, Berlin, New York: Springer-Verlag, pp. 207239, ISBN 978-3-540-08770-0, MR 541023

Seshadri, C. S. (2007), Introduction to the theory of standard monomials, Texts and Readings in Mathematics46, New Delhi: Hindustan Book Agency, ISBN 9788185931784, MR 2347272

Seshadri, C. S. (2012), Standard monomial theory a historical account, Collected papers of C. S. Seshadri.Volume 2. Schubert geometry and representation theory., New Delhi: Hindustan Book Agency, pp. 350, ISBN9789380250175, MR 2905898

32 CHAPTER 9. STANDARD MONOMIAL THEORY

Young, Alfred (1928), On Quantitative Substitutional Analysis, Proc. London Math. Soc. 28 (1): 255292,doi:10.1112/plms/s2-28.1.255

9.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 33

9.5 Text and image sources, contributors, and licenses9.5.1 Text

Newtons identities Source: https://en.wikipedia.org/wiki/Newton{}s_identities?oldid=667597001Contributors: Michael Hardy, CharlesMatthews, Chuunen Baka, Giftlite, Icairns, Rich Farmbrough, Qutezuce, Paul August, Zaslav, Gauge, Reinyday, LutzL, Tony Sidaway, Kb-dank71, Rjwilmsi, Ligulem, Mathbot, JYOuyang, Hillman, KSmrq, Edinborgarstefan, That Guy, From That Show!, SmackBot, Mhym,Druseltal2005, DA3N, Makyen, Simon12, A. Pichler, Thijs!bot, Wikid77, Dogaroon, Headbomb, AntiVandalBot, T0, Shlomi Hillel,Salgueiro~enwiki, David Eppstein, Plasticup, Quantling, Jszigeti, Mulanhua, AlleborgoBot, JerroldPease-Atlanta, Prohlep, Mild Bill Hic-cup, Razorame, Marc van Leeuwen, Addbot, DOI bot, Lightbot, Andreasmperu, AnomieBOT, Citation bot, Citation bot 1, BertSeghers,EmausBot, John of Reading, ZroBot, Tttfkkk, KlappCK, BG19bot, Boriaj, Paolo Lipparini, Brad7777, Rongator, Teddyktchan, SoSivrand Anonymous: 32

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Osculant Source: https://en.wikipedia.org/wiki/Osculant?oldid=548073772 Contributors: R.e.b. and David Eppstein Perpetuant Source: https://en.wikipedia.org/wiki/Perpetuant?oldid=627025070 Contributors: Michael Hardy, Rjwilmsi, R.e.b. and

Trappist the monk Polynomial ring Source: https://en.wikipedia.org/wiki/Polynomial_ring?oldid=670385454 Contributors: Michael Hardy, TakuyaMu-

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Newtons identitiesMathematical statement Formulation in terms of symmetric polynomials Application to the roots of a polynomial Application to the characteristic polynomial of a matrix Relation with Galois theory

Related identities A variant using complete homogeneous symmetric polynomials Expressing elementary symmetric polynomials in terms of power sums Expressing complete homogeneous symmetric polynomials in terms of power sums Expressing power sums in terms of elementary symmetric polynomials Expressing power sums in terms of complete homogeneous symmetric polynomials Expressions as determinants

Derivation of the identities From the special case n = k Comparing coefficients in series As a telescopic sum of symmetric function identities

See also ReferencesExternal links

NullformReferences

OsculantReferences

PerpetuantReferences

Polynomial ringThe polynomial ring K[X] Definition Degree of a polynomialProperties of K[X] Modules

Polynomial evaluation The polynomial ring in several variablesPolynomials The polynomial ring Hilberts Nullstellensatz

Properties of the ring extension R R[X] Summary of the results

GeneralizationsInfinitely many variables Generalized exponentsPower seriesNoncommutative polynomial ringsDifferential and skew-polynomial rings

See also References

Quantum invariantList of invariantsSee alsoReferences Further readingExternal links

Quaternary cubicInvariantsSylvester pentahedronSee alsoReferences

QuippianSee alsoReferences

Standard monomial theoryHistoryApplicationsNotesReferencesText and image sources, contributors, and licensesTextImagesContent license