Hilbert’s tenth problem

By M. Murty, Brandon Fodden. 100 Years of Math Milestones, 2019

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This report is a summary of the negative solution of Hilbert’s Tenth Problem, by Julia Robinson, Yuri Matiyasevich, Martin Davis and Hilary Putnam. I relied heavily on the excellent book by Matiyasevich, Matiyasevich (1993) for both understanding the solution, and writing this summary. Hilbert’s Tenth Problem asks whether or not it is decidable by algorithm if an integer polynomial equation has positive valued roots. The roots of integer valued polynomials are subsets definable in the language of the integers equipped with only addition and multiplication, with constants 0 and 1, and without quantifiers. Further, every definable set of this language is either the roots of a set of polynomials, or its negation. Thus, from a model theoretic perspective, this result shows that the definable sets of even very simple models can be very complicated, atleast from a computational point of view. This summary is divided into five parts. The first part involves basic defintions. The second part sketches a proof of the fact that exponentiation is diophantine. This in turn allows us to construct universal diophantine equations, which will be the subjects of parts three and four. The fifth part will use this to equation to prove the unsolvability of the problem at hand. The final part has some remarks about the entire proof.

Subfields: Theoretical Computer Science

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