A GENERALIZATION OF THE THEOREMS OF CHEVALLEY-WARNING AND AX-KATZ VIA POLYNOMIAL SUBSTITUTIONS

We give conditions under which the number of solutions of a system of polynomial equations over a finite field F-q of characteristic p is divisible by p. Our setup involves the substitution t(i) -> f(t(i)) for auxiliary polynomials f(1), ... , f(n) is an element of F-q[t]. We recover as special cases results of Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving a result that recovers the Ax-Katz theorem. We also consider p-weight degrees, recovering work of Moreno-Moreno, Moreno-Castro, and Castro-Castro-Velez.