Abstract

Igusa’s local zeta function $Z_{f,p}\left(s\right)$ is the generating function that counts the number of integral roots, $N_k\left(f\right)$, of $f\left(x\right)mod{p}^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}\left(p^s\right)$. 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 $N_k\left(f\right)$.

Our proof, when combined with the recent root-counting algorithm of Dwivedi, Mittal and Saxena (Computational Complexity Conference, 2019), yields the first deterministic poly($\left|f\right|,logp$)-time algorithm to compute $Z_{f,p}\left(s\right)$. Previously, an algorithm was known only in the case when $f$ completely splits over ${\mathbb{Q}}_p$; it required the rational roots to use the concept of generating function of a tree (Zúñiga-Galindo, J. Int. Seq., 2003).

Keywords

Mathematical Subject Classification 2010

Milestones

Authors