Abstract
We prove the following statement. Let f ∈ ℝ[x_{1},…,x_{d}], for some d ≥ 3, and assume that f depends nontrivially in each of x_{1},…, x_{d}. Then one of the following holds.

(i)
For every finite sets A_{1},…, 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.
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Raz, O.E., ShemTov, Z. Expanding Polynomials: A Generalization of the ElekesRónyai Theorem to d Variables. Combinatorica 40, 721–748 (2020). https://doi.org/10.1007/s0049302040410
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Mathematics Subject Classification (2010)
 05D99