Kernel Partition Regularity Beyond Linearity

Abstract: A matrix $A$ is called 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 column condition. In this article, we investigate polynomial extensions of Rado's theorem. We exhibit several nonlinear systems of equations that are kernel partition regular and demonstrate that satisfying the column condition continues to ensure kernel partition regularity even when the system is augmented with a polynomial term.