We prove the following statement. Let f ∈ ℝ[ x 1 ,…, x d ], for some d ≥ 3, and assume that f depends non-trivially 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 $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$ | f ( A 1 × … × A d ) | = Ω ( n 3 / 2 ) , with constant of proportionality that depends on deg f . (ii) f is of one of the forms $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right) + \cdots + {p_d}\left( {{x_d}} \right)} \right)$$ f ( x 1 , … , x d ) = h ( p 1 ( x 1 ) + ⋯ + p d ( x d ) ) or $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right) \cdot \ldots \cdot {p_d}\left( {{x_d}} \right)} \right),$$ f ( x 1 , … , x d ) = h ( p 1 ( x 1 ) ⋅ … ⋅ p d ( x d ) ) , 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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展开多项式：Elekes-Rónyai 定理对 d 变量的推广

我们证明以下命题。令 f ∈ ℝ[ x 1 ,…, xd ]，对于某些 d ≥ 3，并假设 f 非平凡地依赖于 x 1 ,…, xd 中的每一个。那么以下之一成立。(i) 对于每个大小为 n 的有限集 A 1 ,…, A d ⊂ℝ，我们有 $$\left| {f\left( {{A_1} \times \ldots \times {A_d}} \right)} \right| = \Omega \left( {{n^{3/2}}} \right),$$ | f ( A 1 × … × A d ) | = Ω (n 3 / 2)，比例常数取决于 deg f。(ii) f 是以下形式之一 $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right ) + \cdots + {p_d}\left( {{x_d}} \right)} \right)$$ f ( x 1 , … , xd ) = h ( p 1 ( x 1 ) +⋯ + pd ( xd ) ) 或 $$f\left( {{x_1}, \ldots ,{x_d}} \right) = h\left( {{p_1}\left( {{x_1}} \right) \cdot \ldots \cdot { p_d}\left( {{x_d}} \right)} \right),$$ f ( x 1 , ... , xd ) = h ( p 1 ( x 1 ) ⋅ … ⋅ pd ( xd ) ) ，对于一些单变量实数多项式 h ( x ), pi ( x ),..., pd ( x )。这概括了 [2,5,7] 的结果，该结果处理了 d = 2 和 d = 3 的情况。