Nonlinear Kernel Partition Regularity: Necessary and Sufficient Conditions · arXivDesk
2407.05542Jul 8, 202417 pages, we have included a necessary condition for the partition regularity of systems of nonlinear equations that are compatible with the classical Rado's column condition, arXiv admin note: text overlap with arXiv:2401.10550
Nonlinear Kernel Partition Regularity: Necessary and Sufficient Conditions
A matrix A is called kernel partition regular if, for every finite coloring of the natural numbers N, there exists a monochromatic solution to the equation AX=0
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. 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 by studying systems of nonlinear equations of the form
AX+P(z)=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.