Partition regularity of polynomial systems near zero

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