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Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

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(2013)cite arxiv:1301.1467.

Abstract

The goal of this article is to prove that there exist at least two classes of nonlinear polynomials that are partition regular on $\N$. To prove this result we introduce a technique based on Nonstandard Analysis and ultrafilters. In synthesis, the technique consists of two facts: the first fact is that, as it is well-known in literature, the partition regularity of specifical polynomials is equivalent to the existence of particular ultrafilters; the second fact, which is showed in this work, is that the existence of such ultrafilters can be studied from the point of view of nonstandard analysis. The core of this second fact is a result, called "Polynomial Bridge Theorem", that entails that the partition regularity of a given polynomial $P(x_1,...,x_n)$ is equivalent to the existence of an ultrafilter $\U$ on $\N$ and elements $\alpha_1,...,\alpha_n$ in the monad of $\U$ (constructed in a properly enlarged hyperextension $^*\N$ of $\N$) such that $P(\alpha_1,...,\alpha_n)=0$.

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