2026-09-30T02:35:12Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25893 2026-09-23T23:53:57Z ddc:004 open_access

Expansion of Trivariate Polynomials Using Proximity Raz, Orit E. Polynomial Expansion Elekes-Rónyai theorem We extend the proximity technique of Solymosi and Zahl [J. Solymosi and J. Zahl, 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 deg(φ). This improves the previous exponent 3/2, due to Raz, Sharir, and De Zeeuw [O. E. Raz M. Sharir and F. de Zeeuw, 2018]. To the best of our knowledge, prior to this work no trivariate polynomial was known to have expansion exceeding Ω(n^{3/2}). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Orit E. Raz 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.86 urn:nbn:de:0030-drops-258936 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.86 eng https://creativecommons.org/licenses/by/4.0/legalcode