Undecidability in number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Listable sets Undecidability in number theory DPRM theorem Consequences of DPRM Prime-producing polynomials Bjorn Poonen Riemann hypothesis Related problems H10 over O k H10 over Q First-order sentences Subrings of Q Rademacher Lecture 1 Status of knowledge November 6, 2017
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 29? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over O k H10 over Q First-order sentences Subrings of Q Status of knowledge
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 29? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems Yes: ( x , y , z ) = (3 , 1 , 1). H10 over O k H10 over Q First-order sentences Subrings of Q Status of knowledge
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 30? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over O k H10 over Q First-order sentences Subrings of Q Status of knowledge
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 30? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over O k H10 over Q Yes: ( x , y , z ) = ( − 283059965 , − 2218888517 , 2220422932). First-order sentences Subrings of Q Status of knowledge (discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant, and M. Beck, following an approach suggested by N. Elkies.)
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Listable sets Do there exist integers x , y , z such that DPRM theorem Consequences of x 3 + y 3 + z 3 = 33? DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over O k H10 over Q First-order sentences Subrings of Q Status of knowledge
Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Listable sets Do there exist integers x , y , z such that DPRM theorem Consequences of x 3 + y 3 + z 3 = 33? DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over O k H10 over Q Unknown. First-order sentences Subrings of Q Status of knowledge
Undecidability in Hilbert’s tenth problem number theory David Hilbert, in the 10th of the list of 23 problems he Bjorn Poonen published after a famous lecture in 1900, asked his audience H10 to find a method that would answer all such questions. Polynomial equations Hilbert’s 10th problem Diophantine sets Hilbert’s tenth problem (H10) Listable sets DPRM theorem Find an algorithm that solves the following problem: Consequences of DPRM Prime-producing input: a multivariable polynomial f ( x 1 , . . . , x n ) with polynomials Riemann hypothesis integer coefficients Related problems output: YES or NO, according to whether H10 over O k H10 over Q there exist integers a 1 , a 2 , . . . , a n such that First-order sentences Subrings of Q f ( a 1 , . . . , a n ) = 0 . Status of knowledge More generally, one could ask for an algorithm for solving a system of polynomial equations, but this would be equivalent, since ⇒ f 2 1 + · · · + f 2 f 1 = · · · = f m = 0 ⇐ m = 0 .
Undecidability in Hilbert’s tenth problem number theory Bjorn Poonen H10 Hilbert’s tenth problem (H10) Polynomial equations Hilbert’s 10th problem Find a Turing machine that solves the following problem: Diophantine sets Listable sets DPRM theorem input: a multivariable polynomial f ( x 1 , . . . , x n ) with Consequences of integer coefficients DPRM Prime-producing polynomials output: YES or NO, according to whether Riemann hypothesis there exist integers a 1 , a 2 , . . . , a n such that Related problems H10 over O k f ( a 1 , . . . , a n ) = 0 . H10 over Q First-order sentences Subrings of Q Status of knowledge Theorem (Davis–Putnam–Robinson 1961 + Matiyasevich 1970) No such algorithm exists! In fact they proved something stronger. . .
Undecidability in Family of polynomial equations number theory Bjorn Poonen . . H10 . Polynomial equations Hilbert’s 10th problem x 2 1 + x 2 2 + x 2 3 + x 2 4 = − 2 NO Diophantine sets Listable sets DPRM theorem x 2 1 + x 2 2 + x 2 3 + x 2 4 = − 1 NO Consequences of DPRM x 2 1 + x 2 2 + x 2 3 + x 2 4 = 0 YES Prime-producing polynomials x 2 1 + x 2 2 + x 2 3 + x 2 Riemann hypothesis 4 = 1 YES Related problems x 2 1 + x 2 2 + x 2 3 + x 2 4 = 2 YES H10 over O k H10 over Q First-order sentences . . Subrings of Q . Status of knowledge The set of a ∈ Z such that x 2 1 + x 2 2 + x 2 3 + x 2 4 = a has a solution in integers is { 0 , 1 , 2 , . . . } =: N
Undecidability in Family of polynomial equations number theory Bjorn Poonen . . H10 . Polynomial equations Hilbert’s 10th problem x 2 1 + x 2 2 + x 2 3 + x 2 4 = − 2 NO Diophantine sets Listable sets DPRM theorem x 2 1 + x 2 2 + x 2 3 + x 2 4 = − 1 NO Consequences of DPRM x 2 1 + x 2 2 + x 2 3 + x 2 4 = 0 YES Prime-producing polynomials x 2 1 + x 2 2 + x 2 3 + x 2 Riemann hypothesis 4 = 1 YES Related problems x 2 1 + x 2 2 + x 2 3 + x 2 4 = 2 YES H10 over O k H10 over Q First-order sentences . . Subrings of Q . Status of knowledge The set of a ∈ Z such that x 2 1 + x 2 2 + x 2 3 + x 2 4 − a = 0 has a solution in integers is { 0 , 1 , 2 , . . . } =: N
Undecidability in Diophantine sets number theory Bjorn Poonen H10 Definition Polynomial equations Hilbert’s 10th problem A ⊆ Z is diophantine if there exists Diophantine sets Listable sets DPRM theorem p ( t , � x ) ∈ Z [ t , x 1 , . . . , x m ] Consequences of DPRM Prime-producing polynomials such that Riemann hypothesis Related problems x ∈ Z m } . H10 over O k A = { a ∈ Z : p ( a , � x ) = 0 has a solution � H10 over Q First-order sentences Subrings of Q Status of knowledge Example The subset N := { 0 , 1 , 2 , . . . } of Z is diophantine, since for a ∈ Z , ( ∃ x 1 , x 2 , x 3 , x 4 ∈ Z ) x 2 1 + x 2 2 + x 2 3 + x 2 a ∈ N ⇐ ⇒ 4 − a = 0 .
Undecidability in Listable sets number theory Bjorn Poonen H10 Polynomial equations Definition Hilbert’s 10th problem Diophantine sets A ⊆ Z is listable if there is a Turing machine Listable sets DPRM theorem such that A is the set of integers that it prints out Consequences of DPRM when left running forever. Prime-producing polynomials Riemann hypothesis Example Related problems H10 over O k The set of integers expressible as a sum of three cubes is H10 over Q First-order sentences Subrings of Q listable. Status of knowledge (Print out x 3 + y 3 + z 3 for all | x | , | y | , | z | ≤ 10 , then print out x 3 + y 3 + z 3 for | x | , | y | , | z | ≤ 100 , and so on.)
Undecidability in Halting problem number theory Bjorn Poonen Can one write a debugger to solve the halting problem? H10 input: program p and natural number x Polynomial equations Hilbert’s 10th problem output: YES if p on input x eventually halts, Diophantine sets Listable sets NO if it enters an infinite loop. DPRM theorem Consequences of DPRM Theorem (Turing 1936) Prime-producing polynomials Riemann hypothesis No such debugger exists; the halting problem is unsolvable. Related problems H10 over O k Sketch of proof: H10 over Q First-order sentences Enumerate all programs. If we had a debugger, we could use Subrings of Q Status of knowledge it to write a new program H such that for every x ∈ N , H on input x halts ⇐ ⇒ program x on input x does not halt. Taking x = H , we find a contradiction: H on input H halts ⇐ ⇒ H on input H does not halt. �
Undecidability in Negative answer to H10 number theory Bjorn Poonen What Davis-Putnam-Robinson-Matiyasevich really proved is: H10 Polynomial equations Hilbert’s 10th problem DPRM theorem: Diophantine ⇐ ⇒ listable Diophantine sets Listable sets DPRM theorem (They showed that the theory of diophantine equations is rich Consequences of DPRM enough to simulate any computer!) Prime-producing polynomials Riemann hypothesis Related problems The DPRM theorem implies a negative answer to H10: H10 over O k H10 over Q First-order sentences The unsolvability of the halting problem provides a Subrings of Q Status of knowledge listable set for which no algorithm can decide membership. So there exists a diophantine set for which no algorithm can decide membership. Thus H10 has a negative answer.
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