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DOI: 10.4230/LIPIcs.SoCG.2026.59

Improved Bound for the k-Variate Elekes-Rónyai Theorem

Authors: Yaara Jahn and Orit E. Raz

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

Let f ∈ ℝ[x₁,…,x_k], for k ≥ 2. For any finite sets A₁,…,A_k ⊂ ℝ, consider the set f(A₁,…,A_k): = {f(a₁,…,a_k)∣ (a₁,⋯,a_k) ∈ A₁×⋯× A_k}, that is, the image of A₁×⋯×A_k under f. Extending a theorem of Elekes and Rónyai, which deals with the case k = 2, and the result of Raz, Sharir, and De Zeeuw [Raz et al., 2018], dealing with the case k = 3, it is proved in Raz and Shem Tov [Raz and Shem{-}Tov, 2020], that for every choice of finite A₁,…, A_k ⊂ ℝ, each of size n, one has (1) |f(A₁,…,A_k)| = Ω(n^{3/2}), unless f has some degenerate special form. In this paper, we introduce the notion of a rank of a k-variate polynomial f, denoted as rank(f). Letting r = rank(f), we prove that (2) |f(A₁,…,A_k)| = Ω(n^{(5r-4)/2r-ε}) , for every ε > 0, where the constant of proportionality depends on ε and on deg(f). This improves the lower bound (1), for polynomials f for which rank(f) ≥ 3. We present an application of our main result, to lower bound the number of distinct d-volumes spanned by (d+1)-tuples of points lying on the moment curve in ℝ^d.

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Yaara Jahn and Orit E. Raz. Improved Bound for the k-Variate Elekes-Rónyai Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jahn_et_al:LIPIcs.SoCG.2026.59,
  author =	{Jahn, Yaara and Raz, Orit E.},
  title =	{{Improved Bound for the k-Variate Elekes-R\'{o}nyai Theorem}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{59:1--59:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.59},
  URN =		{urn:nbn:de:0030-drops-258663},
  doi =		{10.4230/LIPIcs.SoCG.2026.59},
  annote =	{Keywords: Polynomial Expansion, Elekes-R\'{o}nyai theorem}
}

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Improved Bound for the k-Variate Elekes-Rónyai Theorem

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