发表机构
Johann Radon Institute for Computational and Applied Mathematics（约翰·拉登计算与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究埃莱克斯 - 萨博定理中从多项式识别特定群结构难题，给出局部几何准则检测相关超曲面，据此构建边界簇几何框架，应用于超平面与超球面配置及推广的 pinned 距离问题。
AI 中文摘要
埃莱克斯 - 萨博定理表明不可约代数超曲面\(Z(F)\)含很少格点，除非有特定群相关结构。从多项式\(F\)识别此结构具挑战性。给出局部几何准则检测此类超曲面，为边界簇构建几何框架，表明相关边界簇含于坐标切片，给出几何约束。应用于\(\mathbb{R}^d\)中配置及推广 pinned 距离问题。
英文摘要
The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface $Z(F)$ contains few grid points unless it exhibits a specific group-related structure. Identifying this structure from the polynomial $F$ is a challenging problem in combinatorial geometry. Our first main result (Theorem 2.3) provides a local geometric criterion to detect such group-related hypersurfaces. By applying this criterion, we develop a geometric framework for boundary varieties, which are defined by the vanishing of partial derivatives along $Z(F)$. In Theorem 2.4, we show that for group-related varieties, these boundary varieties must be contained in coordinate slices. This gives a strict geometric constraint on the loci where $Z(F)$ becomes tangent to coordinate directions. As an application, we study configurations formed by $d$ coordinate-grid hyperplane families together with a one-parameter polynomial family of hyperspheres in $\mathbb{R}^d$. If one chooses $n$ members from each of these $d+1$ families and obtains $Ω(n^{d-η})$ common incidence points, then the hypersphere family is forced to have a very restricted form: it is concentric in dimensions $d \geq 3$, and in dimension $2$ it is either concentric or consists of fixed-radius circles whose centres lie on a line parallel to a coordinate axis. We also generalize the pinned distance problem initiated by Elekes and Szabó for three points in the plane to $d+1$ points in $\mathbb{R}^d$. More precisely, in Theorem 2.8 we prove that if $d+1$ families of hyperspheres centred at fixed points determine $Ω(n^{d-η})$ points, each lying on one hypersphere from each family, then the centres must be affinely dependent.
