Partition regularity of polynomial systems near zero

Lorenzo Luperi Baglini

doi:10.1007/s00233-021-10179-w

Abstract

Recently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups (S,+) of (mathbb {R}^{+},+). We extend their results to a much larger class of systems whilst weakening the requests on S, using solely basic results about ultrafilters.

Highlights

The study of the partition regularity of linear systems of equations started in the early twentieth century, with the works of Schur, van der Waerden and reads as follows: Theorem 2 (Rado)
Minor modifications of our proofs would work for these strengthened notions, we prefer to use the basic definition of partition regularity so not to have to handle unnecessary complications, since our goal is to show a method to prove the partition regularity near zero of systems that are partition regular on R or Q. 3 This is a particular instance of Theorem 2.1 in [8]; we present a proof here as it is very simple
5 In [9] we showed the partition regularity of polynomials belonging to a much larger class C but, since the definition of C is rather involved, we limit ourself here to present a simpler generalization of Theorem 4

Summary

Introduction

The study of the partition regularity of linear systems of equations started in the early twentieth century, with the works of Schur, van der Waerden and Rado. As we are interested in the notion of partition regularity near zero, and for reasons that will be made precise in Proposition 2, we will often talk about ultrafilters in the set 0+(S), that is defined as follows9.

Objectives

Results

Conclusion