On the Poles of Real Archimedean Zeta Functions

TL;DR

This paper investigates the poles of the real Archimedean zeta function for weighted homogeneous polynomials with isolated singularities, providing explicit conditions for roots of the Bernstein-Sato polynomial to be poles.

Contribution

It derives the meromorphic continuation of the real Archimedean zeta function and establishes a necessary and sufficient condition for roots to be poles, highlighting differences from the complex case.

Findings

Explicit expression for the meromorphic continuation of Z_{f,ϕ}

Necessary and sufficient condition for roots to be poles

Identification of cases where roots do not correspond to poles

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 $Z_{f,\varphi}$ 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 $Z_{f,\varphi}$. 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.