Expansion of trivariate polynomials using proximity

2025/10/14 by Raz, Orit E.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.12191

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+(φ(x)-z)², where φ(x)∈ ℝ[x] has degree at least 3. Then, for every finite A,B,C⊂ ℝ each of size n, one has |f(A,B,C)|=Ω(n^5/3-ε), for every ε>0, where the constant of proportionality depends on ε and on \rm deg(φ). 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).

Citations

• Dense graphs have rigid parts
• Expanding polynomials: A generalization of the Elekes-R 'onyai theorem\n to d variables
• The Elekes-Szabó Theorem in four dimensions
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

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