Elekes-Rónyai型问题综述
A survey of Elekes-Rónyai-type problems
摘要：
我们给出了一个概述，最近的进展围绕一个问题介绍了Elekes和Rónyai。原型问题是要证明多项式$f\in \mathbb{R}[x,y]$在笛卡儿积$A\times B\subset \mathbb{R}^2$上有一个大的像，除非$f$有一个与群相关的特殊形式。我们讨论了一些变体和推广。这包括Elekes-Szabó问题，它将Elekes-Rónyai问题推广到一个关于代数曲面与笛卡尔积相交的上界的问题，以及曲线变体，我们对代数曲线的有限子集的笛卡儿积提出相同的问题。这些问题位于组合学、代数学和几何学的交叉点：它们提出关于代数对象的组合问题，其答案被证明可以应用于涉及距离、直线和圆等基本对象的几何问题，以及加法组合学的和积型问题。作为最近激增的代数技术在组合几何的一部分，一些定量和定性的步骤已经在这个框架内。尽管如此，许多诱人的开放问题仍然存在。
英文摘要：
We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves. These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.
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