Expansion properties of polynomials over finite fields

We establish expansion properties for suitably generic polynomials of degree $d$ in $d+1$ variables over finite fields. In particular, we show that if $P\in {\mathbb{F}}_q[x_1,\dots ,x_{d+1}]$ is a polynomial of degree $d$ coming from an explicit, Zariski dense set, and $X_1,\dots ,X_{d+1}\subseteq {\mathbb{F}}_q$ are suitably large, then $\mathrm{\mid }P(X_1,\dots ,X_{d+1})\mathrm{\mid }=q-O(1)$. Our methods rely on a higher-degree extension of a result of Vinh on point–line incidences over a finite field.