Computing Igusa’s local zeta function of univariates in deterministic polynomial-time

Abstract

Igusa’s local zeta function Z f,p(s) is the generating function that counts the number of integral roots, Nk ( f ), of f (x) mod p k , for all k. It is a famous result, in analytic number theory, that Z f,p is a rational function in Q(p s ). We give an elementary proof of this fact for a univariate polynomial f . Our proof is constructive as it gives a closed-form expression for the number of roots Nk ( f ).