The Elekes-Szab\'o Theorem in four dimensions

TL;DR

This paper extends the Elekes-Szabó theorem to four dimensions, providing bounds on polynomial zeroes over finite sets and applications to geometric configurations.

Contribution

It generalizes the Elekes-Szabó theorem to four dimensions and introduces new bounds for polynomial expansions and geometric incidences.

Findings

Bound of O(n^{8/3}) for polynomial zeroes in four dimensions

Application to bounds on coplanar quadruples on space curves

Application to four-point circles on plane curves

Abstract

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times C\times D$, unless $F$ has a special group-related form. A similar statement holds for $A,B,C,D$ of unequal sizes. This is a four-dimensional extension of our recent improved analysis of the original Elekes-Szab\'o theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.