Partition regularity of polynomial systems near zero

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.

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