Multivariate Polynomial Values in Difference Sets Article Swipe

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Combinatorics Polynomial Mathematics Variety (cybernetics) Degree (music) Algebraic number Multivariate statistics Physics Geometry Mathematical analysis Statistics Acoustics

John R. Doyle , Alex Rice ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2006.15400 · OA: W3037658626

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 $|A|\ll_h Ne^{-c(\log N)^μ}$, where $c=c(h)>0$, $μ=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $μ=1/2$ 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.