An Algebraic Proof of the Polynomial Version of van der Waerden's Theorem

TL;DR

This paper offers an alternative algebraic proof of the polynomial van der Waerden's theorem, traditionally proved via dynamical systems, by utilizing Stone-ech compactification and symbolic polynomials.

Contribution

It introduces symbolic polynomials and employs Stone-ech compactification to provide a new algebraic proof of the polynomial van der Waerden's theorem.

Findings

Proof relies solely on algebraic methods.

Highlights the role of Stone-ech compactification.

Provides an alternative to dynamical systems approach.

Abstract

The polynomial version of van der Waerden's theorem, proved using dynamical systems by V. Bergelson and A. Leibman in 1996, \cite{Bergelson1996}, significantly highlighted the role of dynamical systems in addressing problems related to monochromatic configurations within algebraic structures. In this paper, by introducing symbolic polynomials, we aim to provide an alternative proof of the polynomial version of van der Waerden's theorem relying solely on Stone-\v{C}ech compactification of an infinite discrete semigroup.