2014/08/14 by Hui June Zhu, Zhu, Hui June
Mathematics · #11G #14G #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G #msc:14G
paper · pdf · doi:10.48550/arxiv.1408.3224
11 pages
arxiv created 2015/02/09 · arxiv updated 2015/02/10
The well-known theorem of Ax and Katz gives a p-divisibility 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 over a number field 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 SpecZ. Moreover we show that if G has certain combinatorial conditional number nonzero then the above bound is achieved at special fiber at p for all but finitely many primes p.