Expanding Polynomials: A Generalization of the Elekes-Rónyai Theorem to d Variables

Abstract

We prove the following statement. Let f ∈ ℝ[x1,…,xd], for some d ≥ 3, and assume that f depends non-trivially in each of x1,…, xd. Then one of the following holds.

  1. (i)

    For every finite sets A1,…, Ad ⊂ℝ, each of size n, we have

    $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$

    with constant of proportionality that depends on deg f.

  2. (ii)

    f is of one of the forms

    $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right) + \cdots + {p_d}\left( {{x_d}} \right)} \right)$$

    or

    $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right) \cdot \ldots \cdot {p_d}\left( {{x_d}} \right)} \right),$$

    for some univariate real polynomials h(x), pi(x),…,pd(x). This generalizes the results from [2,5,7], which treat the cases d = 2 and d = 3.

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Correspondence to Orit E. Raz.

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Raz, O.E., Shem-Tov, Z. Expanding Polynomials: A Generalization of the Elekes-Rónyai Theorem to d Variables. Combinatorica 40, 721–748 (2020). https://doi.org/10.1007/s00493-020-4041-0

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Mathematics Subject Classification (2010)

  • 05D99