Further results on Hilbert's Tenth Problem

TL;DR

This paper extends the understanding of Hilbert's Tenth Problem by proving the non-existence of algorithms for certain classes of polynomial Diophantine equations over integers, including specific bounds and transformations.

Contribution

It establishes new undecidability results for polynomial Diophantine equations with bounded variables and specific forms, advancing the record on the original HTP over integers.

Findings

No algorithm exists for determining solutions of certain 9-variable polynomials with a nonnegative variable.

No algorithm can decide solvability of 11-variable polynomials over integers.

Existence of a polynomial encoding prime numbers within its values.

Abstract

Hilbert's Tenth Problem (HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring $\mathbb Z$ of the integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over $\mathbb Z$. We prove that there is no algorithm to determine for any $P(z_1,\ldots,z_9)\in\mathbb Z[z_1,\ldots,z_9]$ whether the equation $P(z_1,\ldots,z_9)=0$ has integral solutions with $z_9\ge0$. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation $P(z_1,\ldots,z_{11})=0$ (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over $\mathbb Z$. We also prove that there is no algorithm to test for any $P(z_1,\ldots,z_{17})\in\mathbb Z[z_1,\ldots,z_{17}]$ whether $P(z_1^2,\ldots,z_{17}^2)=0$ has integral solutions, and that there is a polynomial $Q(z_1,\ldots,z_{20})\in\mathbb Z[z_1,\ldots,z_{20}]$ such that $$\{Q(z_1^2,\ldots,z_{20}^2):\ z_1,\ldots,z_{20}\in\mathbb Z\}\cap\{0,1,2,\ldots\}$$ coincides with the set of all primes.