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arXiv · 2504.17687

The Igusa Zeta function of restricted power series over $\mathbb{Q}_p$

Leonie Dausy
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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 $\Gamma(f)$.

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Leonie Dausy. 2025-04-24. The Igusa Zeta function of restricted power series over $\mathbb{Q}_p$. https://arxiv.org/abs/2504.17687

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