On a theorem of Ax and Katz

The well-known theorem of Ax and Katz gives a pdivisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. It was strengthened by Adolphson–Sperber in terms of Newton polytope of the support set G of V. In this paper we prove that for every generic algebraic variety V over Ǭ supported on G the Adolphson– Sperber bound can be achieved on special fibre at p for a set of prime p of positive density in Spec(Z). Moreover, we show that if an explicitly computable combinatorial function on G is nonzero then the above bound is achieved at special fibre at p for all large enough p.