Polynomial extension of Van der Waerden's Theorem near zero

TL;DR

This paper extends Van der Waerden's theorem to polynomial configurations near zero within dense subrings of real numbers, demonstrating that certain polynomial patterns must appear in any finite partition under specified conditions.

Contribution

It introduces a polynomial version of Van der Waerden's theorem near zero for dense subrings of real numbers, establishing the existence of polynomial configurations in finite partitions.

Findings

Polynomial configurations exist near zero in dense subrings.

The theorem applies to polynomials with zero constant term and positive near zero.

It guarantees the presence of polynomial patterns in any finite partition.

Abstract

Let $S$ be a dense subring of the real numbers. In this paper we prove a polynomial version of Van der Waerden's theorem near zero. In fact, we prove that if $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and there exists $\delta > 0$ such that $p_i(x) > 0$ for every $x \in (0,\delta)$ and for every $i=1,\ldots , m$. Then for any finite partition $\mathcal{C}$ of \( S\cap(0,1) \) and every sequence $f:\mathbb{N}\to S\cap(0,1)$ satisfying $\sum_{n=1}^\infty f(n)<\infty$, there exist a cell $C \in \mathcal{C}$, an element $a \in S$, and $F \in P_f(\mathbb{N})$ such that \[ \{ a + p_i(\sum_{t \in F} f(t)) : i = 1,2,\ldots,m \} \subseteq C. \]