Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

TL;DR

This paper proves that determining partition regularity of polynomial equations over integers and certain fields is undecidable, linking it to Hilbert's tenth problem and analyzing the complexity of related sets.

Contribution

It establishes undecidability results for polynomial partition regularity over various domains, conditional on Hilbert's tenth problem, and determines the complexity of these sets.

Findings

Partition regularity sets are undecidable over integers, conditional on Hilbert's tenth problem.

The complexity of polynomial sets is characterized as ^0-complete.

Introduces principles for density Ramsey theory and measure-preserving systems in algebraic structures.

Abstract

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $\Pi_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.