Partition regularity of polynomial systems near zero

TL;DR

This paper extends the results on partition regularity of polynomial systems near zero to a broader class of systems and semigroups, using ultrafilter techniques to weaken initial assumptions.

Contribution

It generalizes previous findings by broadening the class of polynomial systems and semigroups for which partition regularity near zero holds, employing ultrafilter methods.

Findings

Extended partition regularity results to more systems

Weakened conditions on the dense subsemigroups

Utilized ultrafilter techniques for proofs

Abstract

Recently, S.~Kanti Patra and Md.~Moid Shaik proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups $S$ of $((0,\infty),+)$. We extend their results to a much larger class of systems whilst weakening the requests on $S$, using solely basic results about ultrafilters.