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

06/16/2020

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by   Ashish Dwivedi, et al.

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Igusa's local zeta function Z_f,p(s) is the generating function that counts the number of integral roots, N_k(f), of f(𝐱) 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 N_k(f). Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly(|f|, log p) time algorithm to compute Z_f,p(s). Previously, an algorithm was known only in the case when f completely splits over ℚ_p; it required the rational roots to use the concept of generating function of a tree (Zúñiga-Galindo, J.Int.Seq., 2003).