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Expansion of trivariate polynomials using proximity

Orit E. Raz

TL;DR

The paper tackles the problem of expansion for trivariate polynomials on Cartesian products, extending the proximity method of Solymosi and Zahl to three variables. It proves that for $f(x,y,z)=(x-y)^2+((x)-z)^2$ with $\deg\varphi\ge 3$, the image size on $A\times B\times C$ with $|A|=|B|=|C|=n$ satisfies $|f(A,B,C)|=\Omega(n^{5/3-\varepsilon})$ for every $\varepsilon>0$, improving the prior $\Omega(n^{3/2})$ bound. The approach blends a trivariate proximity argument with a plane-curve incidence framework (including a $s$-dimensional curve-family bound and symmetry considerations) to tightly control degeneracies. This establishes a nontrivial subquadratic expansion for a concrete trivariate family and lays groundwork for extending the method to wider classes of trivariate polynomials.

Abstract

We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(\varphi(x)-z)^2$, where $\varphi(x)\in \mathbb{R}[x]$ has degree at least 3. Then, for every finite $A,B,C\subset \mathbb{R}$ each of size $n$, one has $|f(A,B,C)|=Ω(n^{5/3-\varepsilon})$, for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(\varphi)$. This improves the previous exponent $3/2$, due to Raz, Sharir, and De Zeeuw [Israel J. Math. (2018)]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding $Ω(n^{3/2})$.

Expansion of trivariate polynomials using proximity

TL;DR

The paper tackles the problem of expansion for trivariate polynomials on Cartesian products, extending the proximity method of Solymosi and Zahl to three variables. It proves that for with , the image size on with satisfies for every , improving the prior bound. The approach blends a trivariate proximity argument with a plane-curve incidence framework (including a -dimensional curve-family bound and symmetry considerations) to tightly control degeneracies. This establishes a nontrivial subquadratic expansion for a concrete trivariate family and lays groundwork for extending the method to wider classes of trivariate polynomials.

Abstract

We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let , where has degree at least 3. Then, for every finite each of size , one has , for every , where the constant of proportionality depends on and on . This improves the previous exponent , due to Raz, Sharir, and De Zeeuw [Israel J. Math. (2018)]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding .
Paper Structure (6 sections, 7 theorems, 60 equations)

This paper contains 6 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Point-curve incidences in the plane
  4. Symmetry of curves
  5. Complete bipartite graphs
  6. Proof of Theorem \ref{['main']}

Key Result

Theorem 1

Let $k\ge 3$ and assume that $f\in \mathbb{R}[x_1,\ldots,x_k]$ depends non-trivially on each of its varaiables. Then one of the following holds: (i) For every finite $A_1,\ldots,A_k\subset \mathbb{R}$, with $|A_i|=n$, for $i=1,\ldots,k$, one has where the constant of proportionality depends only on $\deg(f)$ and on $k$. (ii) $f$ is of one of the special forms for some univariate real polynomials

Theorems & Definitions (10)

  • Theorem 1: RazShaDeZee4DRazShe
  • Theorem 2
  • Theorem 3: Sharir--Zahl ShaZah
  • Lemma 4: Pach and De Zeeuw PachDeZeeuw
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof