发表机构
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences; Department of Mathematics University of Mississippi; Department of Mathematics and Statistics Loyola University Chicago（奥地利科学院计算与应用数学约翰·拉德森研究所; 密西西比大学数学系; 芝加哥洛约拉大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在函数域$\mathbb{F}_q[t]$中推广Sárközy定理至多变量方程，并利用简单观察移除Green论证中的技术性条件，扩展了Li和Sauermann的结果。
AI 中文摘要
Sárközy定理指出，若$A \subset \mathbb{Z}$具有正的上渐近密度，则存在不同的$a_1, a_2 \in A$以及$n \in \mathbb{Z}$，使得$a_1-a_2 = n^2$。若将$n^2$替换为任意常数项为零的多项式$F \in \mathbb{Z}[x]$的$F(n)$，同样的结论依然成立。Green证明了Sárközy定理在$\mathbb{F}_q[t]$上的类比，并给出了强定量界，但要求多项式$F \in \mathbb{F}_q[x]$的根数满足一个技术性条件。该条件最近被Li和Sauermann移除。在本文中，我们推广Green的论证，以处理$\mathbb{F}_q[t]$中更多变量的方程，同时指出该技术性条件可通过一个简单观察加以移除。
英文摘要
Sárközy's theorem says that if $A \subset \mathbb{Z}$ has positive upper asymptotic density, then there are distinct $a_1, a_2 \in A$ and $n \in \mathbb{Z}$ such that $a_1-a_2 = n^2$. The same is true if $n^2$ is replaced by $F(n)$ for any polynomial $F \in \mathbb{Z}[x]$ with constant term zero. Green proved an $\mathbb{F}_q[t]$-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial $F \in \mathbb{F}_q[x]$. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in $\mathbb{F}_q[t]$, while pointing out that the technical condition can be removed by means of a simple observation.
