undecidability in number theoryggranja/ta... · hilbert’s 10th problem diophantine sets listable...
TRANSCRIPT
![Page 1: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/1.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Undecidability in number theory
Bjorn Poonen
MIT
Novos Talentos em MatematicaLisboa
July 15, 2010
![Page 2: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/2.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 29?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 3: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/3.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 29?
Yes: (x , y , z) = (3, 1, 1).
Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 4: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/4.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 30?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 5: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/5.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 30?
Yes: (x , y , z) = (3, 1, 1).
Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).
Unknown.
(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 6: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/6.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 33?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 7: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/7.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 33?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).
Unknown.
(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 8: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/8.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 47?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 9: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/9.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 51?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 10: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/10.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 84?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 11: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/11.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 54?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 12: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/12.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 11?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 13: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/13.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = −98?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 14: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/14.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 427?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 15: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/15.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 689?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 16: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/16.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 138?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 17: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/17.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 823?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 18: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/18.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 549?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 19: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/19.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = −190?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 20: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/20.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 3837?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 21: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/21.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 3992?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 22: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/22.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 5566?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 23: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/23.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 5172?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 24: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/24.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 9572?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 25: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/25.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 17260?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
![Page 26: Undecidability in number theoryggranja/Ta... · Hilbert’s 10th problem Diophantine sets Listable sets DPRM theorem Consequences of DPRM Prime-producing polynomials Riemann hypothesis](https://reader034.vdocuments.us/reader034/viewer/2022051808/600e99285e1e2423bb3dc5ab/html5/thumbnails/26.jpg)
Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 57227?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 27491?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = −72888?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 221947?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 722900?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 828376?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = −372349?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 2958484?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 9598772?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 17838782?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 70871236846?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x3 + y3 + z3 = 9798701987509879873490579790709798?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x17 + y17 + z17 = 17?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
x34 + x5y23 + z17 + xyz = 196884?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Examples of polynomial equations
Do there exist integers x , y , z such that
536x287196896 − 210y287196896 + 777x3y16z4732987
−1111x54987896 − 2823y927396 + 27x94572y9927z999
−936718x726896 + 887236y726896 − 9x24572y7827z13
+89790876x26896 + 30y26896 + 987x245y6z6876
+9823709709790790x28 − 1987y28 + 1467890461986x2y6z4
+80398600x2z12 − 27980186xy + 3789720156y2 + 9328769x
−1956820y − 27589324985727098790768645846898z = 389?
Yes: (x , y , z) = (3, 1, 1).Yes: (x , y , z) = (−283059965,−2218888517, 2220422932).Unknown.(discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant,and M. Beck, following an approach suggested by N. Elkies.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Hilbert’s tenth problem
D. Hilbert, in the 10th of the list of 23 problems hepublished after a famous lecture in 1900, asked his audienceto find a method that would answer all such questions.
Hilbert’s tenth problem (H10)
Find an algorithm that solves the following problem:
input: a multivariable polynomial f (x1, . . . , xn) withinteger coefficients
output: YES or NO, according to whether there existintegers a1, a2, . . . , an such thatf (a1, . . . , an) = 0.
More generally, one could ask for an algorithm for solving asystem of polynomial equations, but this would beequivalent, since
f1 = · · · = fm = 0 ⇐⇒ f 21 + · · ·+ f 2
m = 0.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Hilbert’s tenth problem
Hilbert’s tenth problem (H10)
Find a Turing machine that solves the following problem:
input: a multivariable polynomial f (x1, . . . , xn) withinteger coefficients
output: YES or NO, according to whether there existintegers a1, a2, . . . , an such thatf (a1, . . . , an) = 0.
Theorem (Davis-Putnam-Robinson 1961 +Matiyasevich 1970)
No such algorithm exists.
In fact they proved something stronger. . .
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Polynomial families of polynomial equations
Example
Starting with
x3 + y3 + 2tx2 + (t + 10)xy − 7 = 0
we can get infinitely many polynomial equations in x and yby substituting particular integers for t:
t = 1: x3 + y3 + 2x2 + 11xy − 7 = 0
t = 2: x3 + y3 + 4x2 + 12xy − 7 = 0
t = 3: x3 + y3 + 6x2 + 13xy − 7 = 0
...
For some values of t, it will have a solution in integers x , y ,and for some it will not.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Diophantine sets
Definition
A ⊆ Z is diophantine if there exists
p(t,~x) ∈ Z[t, x1, . . . , xm]
such that
A = { a ∈ Z : (∃~x ∈ Zm) p(a,~x) = 0 }.
Example
The subset N := {0, 1, 2, . . . } of Z is diophantine, since fora ∈ Z,
a ∈ N ⇐⇒ (∃x1, x2, x3, x4 ∈ Z) x21 + x2
2 + x23 + x2
4 − a = 0.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Listable sets
Definition
A ⊆ Z is listable (computably enumerable) if there is aTuring machine such that A is the set of integers that itprints out when left running forever.
Example
The set of integers expressible as a sum of three cubes islistable.
(Print out x3 + y3 + z3 for all |x |, |y |, |z | ≤ 10,then print out x3 + y3 + z3 for |x |, |y |, |z | ≤ 100,and so on.)
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Negative answer to H10
What Davis-Putnam-Robinson-Matiyasevich really proved is:
DPRM theorem: Diophantine ⇐⇒ listable
(They showed that the theory of diophantine equations isrich enough to simulate any computer!)
The DPRM theorem implies a negative answer to H10:
The unsolvability of the Halting Problem provides alistable set for which no algorithm can decidemembership.
So there exists a diophantine set for which no algorithmcan decide membership.
Thus H10 has a negative answer.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
More fun consequences of the DPRM theorem
“Diophantine ⇐⇒ listable” has applications beyond thenegative answer to H10:
Prime-producing polynomials
Diophantine statement of the Riemann hypothesis
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
The set of primes equals the set of positive values assumedby the 26-variable polynomial
(k + 2){1− ([wz + h + j − q]2
+[(gk + 2g + k + 1)(h + j) + h − z ]2
+[16(k + 1)3(k + 2)(n + 1)2 + 1− f 2]2
+[2n + p + q + z − e]2 + [e3(e + 2)(a + 1)2 + 1− o2]2
+[(a2 − 1)y2 + 1− x2]2 + [16r2y4(a2 − 1) + 1− u2]2
+[((a + u2(u2 − a))2 − 1)(n + 4dy)2 + 1− (x + cu)2]2
+[(a2 − 1)`2 + 1−m2]2
+[ai + k + 1− `− i ]2 + [n + ` + v − y ]2
+[p + `(a− n − 1) + b(2an + 2a− n2 − 2n − 2)−m]2
+[q + y(a− p − 1) + s(2ap + 2a− p2 − 2p − 2)− x ]2
+[z + p`(a− p) + t(2ap − p2 − 1)− pm]2)}
as the variables range over nonnegative integers(J. Jones, D. Sato, H. Wada, D. Wiens).
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Riemann hypothesis
The DPRM theorem gives an explicit polynomial equationthat has a solution in integers if and only if the Riemannhypothesis is false.
Sketch of proof.
One can write a computer program that, when leftrunning forever, will detect a counterexample to theRiemann hypothesis if one exists.
Simulate this program with a diophantine equation.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
H10 over Q
It is not known whether there exists an algorithm thatdecides whether a multivariable polynomial equation hasa solution in rational numbers.
If Z is diophantine over Q, then the negative answer forZ implies a negative answer for Q.
But there is a conjecture that implies that Z is notdiophantine over Q:
Conjecture (Mazur 1992)
For any polynomial equation f (x1, . . . , xn) = 0 with rationalcoefficients, if S is the set of rational solutions, then theclosure of S in Rn has at most finitely many connectedcomponents.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
First-order sentences over ZIn terms of logic, H10 asks for an algorithm to decidethe truth of positive existential sentences
(∃x1∃x2 · · · ∃xn) p(x1, . . . , xn) = 0.
in the language of rings, where the variables run overintegers.
More generally, one can ask for an algorithm to decidethe truth of arbitrary first-order sentences, in which anynumber of bound quantifiers ∃ and ∀ are permitted: atypical such sentence is
(∃x)(∀y)(∃z)(∃w) (x · z + 3 = y2) ∨ ¬(z = x + w)
Long before DPRM, the work of Church, Godel, andTuring in the 1930s made it clear that there was noalgorithm to solve the harder problem of deciding thetruth of first-order sentences over Z.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
First-order sentences over Q
Though it is not known whether Z is diophantine (i.e.,definable by a positive existential formula) over Q, we have
Theorem (J. Robinson 1949)
One can characterize Z as the set of t ∈ Q such that aparticular first-order formula of the form
(∀~x)(∃~y)(∀~z)(∃~w) p(t,~x , ~y ,~z , ~w) = 0
is true, when the variables range over rational numbers.
Corollary
There is no algorithm to decide the truth of a first-ordersentence over Q.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Using quaternion algebras, one can improve J. Robinson’sresult to
Theorem (P. 2007)
It is possible to define Z in Q with a formula with 2universal quantifiers followed by 7 existential quantifiers.
Corollary
There is no algorithm for deciding, given an algebraic familyof morphisms of varieties, whether there exists one that issurjective on rational points.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Theorem (P. 2007)
The set Z equals the set of t ∈ Q such that
(∀a, b)(∃x1, x2, x3, x4, y2, y3, y4)
(a + x21 + x2
2 + x23 + x2
4 )(b + x21 + x2
2 + x23 + x2
4 )
·[(
x21 − ax2
2 − bx23 + abx2
4 − 1)2
+2309∏n=0
((n − t − 2x1)
2 − 4ay22 − 4by2
3 + 4aby24 − 4
)2
]= 0
is true, when the variables range over rational numbers.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
H10 over subrings of QLet P = {2, 3, 5, . . .}. There is a bijection
{subsets of P} ↔ {subrings of Q}S 7→ Z[S−1].
Examples:
S = ∅, Z[S−1] = Z, answer is negativeS = P, Z[S−1] = Q, answer is unknown
What happens for S in between?
How large can we make S (in the sense of density) andstill prove a negative answer for H10 over Z[S−1]?
For finite S , a negative answer follows from work ofRobinson, who used the Hasse-Minkowski theorem(local-global principle) for quadratic forms.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
H10 over subrings of Q, continued
Theorem (P., 2003)
There exists a computable set of primes S ⊂ P of density 1such that
1. There exists a curve E such that E (Z[S−1]) is aninfinite discrete subset of E (R). (So the analogue ofMazur’s conjecture for Z[S−1] is false.)
2. H10 over Z[S−1] has a negative answer.
The proof takes E to be an elliptic curve (minus ∞), anduses properties of integral points on elliptic curves.
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Undecidability innumber theory
Bjorn Poonen
H10
Polynomial equations
Hilbert’s 10th problem
Diophantine sets
Listable sets
DPRM theorem
Consequences ofDPRM
Prime-producingpolynomials
Riemann hypothesis
Related problems
H10 over QFirst-order sentences
Subrings of QStatus of knowledge
Ring H10 1st order theory
C YES YES
R YES YES
Fq YES YES
p-adic fields YES YES
Fq((t)) ? ?
number field ? NO
Q ? NO
global function field NO NO
Fq(t) NO NO
C(t) ? ?
C(t1, . . . , tn), n ≥ 2 NO NO
R(t) NO NO
Ok ? NO
Z NO NO