A limiting result for the Ramsey theory of functional equations

TL;DR

This paper characterizes the partition regularity of systems of functional equations, linking it to constant solutions, and applies this to Diophantine, $S$-unit, and polynomial exponential equations.

Contribution

It provides a complete characterization of partition regularity for certain functional and polynomial exponential systems, extending classical results.

Findings

Only the equation x=y is infinitely PR among two-variable Diophantine equations.

PR of S-unit equations is characterized, showing Rado's Theorem fails for certain subgroups.

Complete PR characterization for specific polynomial exponential equations.

Abstract

We study systems of functional equations whose solutions can be parameterized in function of one variable; our main result proves that the partition regularity (PR) of such systems can be completely characterized by the existence of constant solutions. As applications of this result, we prove the following: (1) A complete characterization of the PR of systems of Diophantine equations in two variables over $\mathbb{N}$. In particular, we prove that the only infinitely PR irreducible equation in two variables is $x=y$; (2) PR of $S$-unit equations and the failure of Rado's Theorem for finitely generated multiplicative subgroups of $\mathbb{C}$; and (3) a complete characterization of the PR of two classes of polynomial exponential equations.