论文标题
差异集中的多元多项式值
Multivariate Polynomial Values in Difference Sets
论文作者
论文摘要
对于$ \ ell \ geq 2 $和$ h \ in \ mathbb {z} [x_1，\ dots，x _ {\ ell}] $ $ k \ geq 2 $，我们表明，每个集合$ a \ subseteq \ subseteq \ subseteq \ subseteq \ subseteq \ {1,2，\ dots，n \ dots，n \ n \ n \ n \ n \ n \ n \ n \ n nonyzero差异$ h（\ mathbb {z}^{\ ell}）$满足$ | a | \ ll_h ne^ne^{ - c（\ log n）^μ} $，其中$ c = c（h）> 0 $，$μ= [（k-1）提供的$ h（\ mathbb {z}^{\ ell}）$包含每个自然数的倍数，$ h $满足某些非语言条件。我们还详细探讨了这些条件，利用了代数几何形状的各种工具。
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.
