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
$P(\alpha_1,...,\alpha_n)=0$.
Users
Please
log in to take part in the discussion (add own reviews or comments).