Multivariate Polynomial Values in Difference Sets

TL;DR

This paper establishes bounds on the size of sets avoiding certain polynomial difference patterns, using algebraic geometry to analyze the conditions on the polynomial and its values.

Contribution

It provides new bounds for difference sets avoiding polynomial values, extending previous results to multivariate polynomials with specific algebraic conditions.

Findings

Sets avoiding polynomial difference patterns are small, with size bounds involving exponential decay.

The results depend on the polynomial's degree, number of variables, and algebraic properties.

Algebraic geometry tools are used to analyze the conditions on the polynomial h.

Abstract

For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq 2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log N)^{\mu}}$, where $c=c(h)>0$, $\mu=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $\mu=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in detail, drawing on a variety of tools from algebraic geometry.