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 $P\in F_q[x_1,\dots ,x_{d+1}]$ is a polynomial of degree d, whose coefficients avoid the zero locus of some explicit polynomial of degree $O_d\left(1\right)$, and $X_1,\dots ,X_{d+1}\subseteq F_q$ are suitably large, then $\left|P\right(X_1,\dots ,X_{d+1}\left)\right|=q-O\left(1\right)$. Our methods rely on a higher-degree extension of a result of Vinh on point–line incidences over a finite field.