Rado functionals and applications

TL;DR

This paper investigates Rado functionals and the maximal Rado condition to establish new criteria for the partition regularity of polynomial equations over infinite subsets of rings, extending previous results in the field.

Contribution

It strengthens the maximal Rado condition and applies it to derive new partition regularity conditions for polynomial equations and inequalities.

Findings

Extended partition regularity criteria for polynomial equations.

Provided sufficient conditions for partition regularity over infinite subsets.

Generalized previous results to broader classes of equations.

Abstract

We study Rado functionals and the maximal condition (first introduced by J. M. Barret et al.) in terms of the partition regularity of mixed systems of linear equations and inequalities. By strengthening the maximal Rado condition, we provide a sufficient condition for the partition regularity of polynomial equations over some infinite subsets of a given commutative ring. By applying these results, we derive an extension of a previous result obtained by M. Di Nasso and L. Luperi Baglini concerning partition regular inhomogeneous polynomials in three variables and also conditions for the partition regularity of equations of the form $H(xz^\rho,y)=0$, where $\rho$ is a non-zero rational and $H\in\mathbb{Z}[x,y]$ is a homogeneous polynomial.