On a paucity result in Incidence Geometry

发生几何与有限域上的多项式展开

Incidence geometry and polynomial expansion over finite fields

本文通过谱理论和代数几何方法，在有限域上建立更高次型Szemerédi-Trotter定理，并应用于多项式展开研究。

探索关联几何与和积现象

Exploration on Incidence Geometry and Sum-Product Phenomena

本论文研究Erdős-Szemerédi猜想与关联几何的关系，将实数域上的和积估计技术推广到有限域，并在环Z/p^3Z上获得平分线数量的估计结果。

An incidence conjecture of Bourgain over fields of positive characteristic

In this note we generalize a recent theorem of Guth and Katz on incidences between points and lines in $3$-space from characteristic $0$ to characteristic $p$, and we explain how some of the special features of algebraic geometry in characteristic $p$ manifest themselves in problems of incidence geometry.

Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in na…

有限域上若干值集的下界：关联几何与Bourgain群扩张定理

Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem

本文通过过渡原理和关联几何，证明了若干结构化序列（如阶乘、Pochhammer乘积等）在素域上的值集具有多项式下界，并推广到Möbius变换情形，得到超过平方根尺度的结果。