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

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 $\le D$, 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 $\mathrm{\Gamma }(f)$.