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Expansion of Trivariate Polynomials Using Proximity

Author Orit E. Raz

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LIPIcs.SoCG.2026.86.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Polynomial Expansion
  • Elekes-Rónyai theorem

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Abstract

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}).

Cite As Get BibTex

Orit E. Raz. Expansion of Trivariate Polynomials Using Proximity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.86

Author Details

Orit E. Raz
  • Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva, Israel

References

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  3. J. Pach and F. de Zeeuw. Distinct Distances on Algebraic Curves in the Plane. Combinat., Probab. Comput., 26(1):99-117, 2017. URL: https://doi.org/10.1017/S0963548316000225.
  4. O. E. Raz and Z. Shem-Tov. Expanding polynomials: A genralization of the Eleke-Rónyai theorem to d variables. Combinatorica, 40(5):721-748, 2020. URL: https://doi.org/10.1007/S00493-020-4041-0.
  5. O. E. Raz and J. Solymosi. Dense graphs have rigid parts. Discrete Cumput. Geom., 69:1079-1094, 2023. URL: https://doi.org/10.1007/S00454-022-00477-7.
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  7. O. E. Raz M. Sharir and F. de Zeeuw. The Elekes-Szabó theorem in four dimensions. Israel J. Math., 227:663-690, 2018. Google Scholar
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  10. E. Szemerédi and W. T Trotter. A Combinatorial Distinction Between the Euclidean and Projective Planes. European Journal of Combinatorics, 4:385-394, 1983. URL: https://doi.org/10.1016/S0195-6698(83)80036-5.
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