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 Pq[x₁,…,xd+1] is a polynomial of degree d coming from an explicit, Zariski dense set, and X₁,…,Xd+1q are suitably large, then |P(X₁,…,Xd+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.