The Igusa Zeta function of restricted power series over $\mathbb{Q}_p$
Leonie Dausy
TL;DR
This work investigates whether the Igusa zeta function $Z_f(s)$ of a $p$-adic restricted power series $f$ is determined by a truncation $f_D$ at degree $D$. It provides a counterexample in one variable showing that, in general, truncation does not preserve $Z_f(s)$. However, it establishes a positive invariance result under non-degeneracy: if $f$ and $f'$ do not vanish simultaneously (one-variable case) or more generally if $f$ is non-degenerate over $\ ext{F}_p$ with respect to all faces of its Newton polyhedron, then for sufficiently large $D$, $Z_f(s)=Z_{f_D}(s)$. In the multi-variable setting, the Denef–Hoornaert framework yields a concrete formula for $Z_f(s)$ based on the Newton polyhedron, and the truncation invariance follows because the Newton data stabilizes for large $D$. The results illuminate when truncation suffices to compute Igusa zeta functions of $p$-adic analytic restricted series.
Abstract
In this article, we ask whether the Igusa zeta function of a restricted power series over $\mathbb{Q}_p$ can be determined solely from the terms of degree at most $D$. That is, we ask whether the truncated polynomial $f_D$, consisting of all terms of f of degree $\leq D$, yields the same Igusa zeta function as $f$ for sufficiently large $D$. Our main results include a counterexample already in the one-variable case, but also a positive result under the condition that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $Γ(f)$.