Expansion properties of polynomials over finite fields
Nuno Arala, Sam Chow · Finite Fields and Their Applications · 2025
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 ∈ F q [ x 1 , … , x d + 1 ] is a polynomial of degree d , whose coefficients avoid the zero locus of some explicit polynomial of degree O d ( 1 ) , and X 1 , … , X d + 1 ⊆ F q are suitably large, then | P ( X 1 , … , 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.