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

Abstract

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

• (i) For every finite sets A₁,…, A_d ⊂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.

• (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), p_i(x),…,p_d(x). This generalizes the results from [2,5,7], which treat the cases d = 2 and d = 3.