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We establish expansion properties for suitably generic polynomials of degree d in d+1 variables over finite fields. In particular, we show that if Pqx₁, , x₃+₁ is a polynomial of degree d coming from an explicit, Zariski dense set, and X₁, , X₃+₁q are suitably large, then |P (X₁, , X₃+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.
Arala et al. (Wed,) studied this question.