On polynomial expanders with many variables

Abstract

For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$P(x_1,x_2,\dots ,x_{n+2})=\sum _{i=1}^n(x_2-x_1{)}^{i-1}{x}_1^{n-i}{x}_{i+2}.$$ We prove that, for any finite set $A$ of complex numbers, $$\left|\right\{P(x_1,x_2,\dots ,x_{n+2}):x_i\in A\left\}\right|\gg |A{|}^n.$$ The implicit constant in $\gg$ may depend only on $n.$