On polynomial expanders with many variables

Source abstract

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