On the Poles of Real Archimedean Zeta Functions

Abstract

This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial $f\in \mathbb{R}[x,y]$ with an isolated singularity at the origin. By applying a weighted blow-up, we derive the meromorphic continuation of $Z_{f,\phi }$ to $\text{Re }s>-1$. This explicit expression yields a necessary and sufficient condition for a root $s\in (-1,0)$ of the Bernstein-Sato polynomial $b_f(s)$ to be a pole of $Z_{f,\phi }$. Unlike the complex case established by F. Loeser (1985), this condition may fail in certain obvious cases -- such as when $f$ is odd or even in $x$, $y$, or $(x,y)$ -- so not all such roots necessarily become poles.