A symmetric multivariate Elekes-Rónyai theorem

Abstract

We consider a polynomial $P\in \mathbb{R}[x_1,\cdots ,x_d]$ of degree $\hat{I}´$ that depends non-trivially on each of $x_1,...,x_d$ with $d\ge 2$. For any integer $t$ with $2\le t\le d$, any natural number $n\in \mathbb{N}$, and any finite set $A\subset \mathbb{R}$ of size $n$, our first result shows that \[ |P(A, A, \dots, A)| \gg_δ n^{\frac{3}{2} - \frac{1}{2^{d-t+2}}}, \] unless \begin{align*} &P(x_1, x_2, \dots, x_d) = f\big( u_1(x_1) + u_2(x_2) + \cdots + u_d(x_d) \big) \quad \text{or } &P(x_1, x_2, \dots, x_d) = f\big( v_1(x_1) v_2(x_2) \cdots v_d(x_d) \big), \end{align*} where $f$, $u_i$, and $v_i$ are nonconstant univariate polynomials over $\mathbb{R}$, and there exists an index subset $I\subseteq [d]$ with $|I|=t$ such that for any $i,j\in I$, we have $u_i=\hat{I}{»}_{ij}{u}_j$ (in the additive case) or $|v_i|=|v_j{|}^{\hat{I}{º}_{ij}}$ (in the multiplicative case) for some constants $\hat{I}{»}_{ij}\in {\mathbb{R}}^{\ne 0},\hat{I}{º}_{ij}\in {\mathbb{Q}}^{+}$. This result generalizes the symmetric Elekes-Rónyai theorem proved by Jing, Roy, and Tran. Our second result is a generalized Erdős-Szemerédi theorem for two polynomials in higher dimensions, generalizing another theorem by Jing, Roy, and Tran. A key ingredient in our proofs is a variation of a theorem by Elekes, Nathanson, and Ruzsa.