Nonlinear Kernel Partition Regularity: Necessary and Sufficient Conditions

TL;DR

This paper extends Rado's classical theorem on kernel partition regular matrices to certain nonlinear systems, establishing necessary and sufficient conditions for their partition regularity involving polynomial extensions.

Contribution

It introduces a nonlinear extension of Rado's theorem, providing a structural necessary condition for the partition regularity of nonlinear systems of equations.

Findings

Classical column condition guarantees kernel partition regularity in nonlinear systems.

Several nonlinear systems are shown to be kernel partition regular under the extended conditions.

A new structural necessary condition for nonlinear Rado-type systems is established.

Abstract

A matrix \( A \) is called \emph{kernel partition regular} if, for every finite coloring of the natural numbers \( \mathbb{N} \), there exists a monochromatic solution to the equation \( A\vec{X} = 0 \). In 1933, Rado characterized such matrices by showing that a matrix is kernel partition regular if and only if it satisfies the so-called \emph{column condition}. In this article, we investigate polynomial extensions of Rado's theorem by studying systems of nonlinear equations of the form $A \vec{X} + P(z) = \vec{0},$ where $A$ is a matrix with integer entries and $P$ is a finite set of polynomials in one variable with no constant term. We present several nonlinear systems of equations that are kernel partition regular, showing that the classical column condition still guarantees kernel partition regularity, even when the system is extended by adding a nonlinear polynomial term. We then establish a structural necessary condition for the partition regularity of nonlinear Rado-type systems, extending the classical column condition to a nonlinear setting. This condition generalizes Rado's classical column condition by exploring the dependencies between the linear and higher-degree polynomial components of the system.