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 $P\in\mathbb{F}_q[x_1,\ldots,x_{d+1}]$ is a polynomial of degree $d$ coming from an explicit, Zariski dense set, and $X_1,\ldots,X_{d+1}\subseteq\mathbb{F}_q$ are suitably large, then $|P(X_1,\ldots,X_{d+1})|=q-O(1)$. Our methods rely on a higher-degree extension of a result of Vinh on point--line incidences over a finite field.