Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.07679 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912754197069824 |
|---|---|
| author | Chen, Zhikuang Zuo, Huaiqing |
| author_facet | Chen, Zhikuang Zuo, Huaiqing |
| contents | 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 $Z_{f,φ}$ 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 $Z_{f,φ}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Poles of Real Archimedean Zeta Functions Chen, Zhikuang Zuo, Huaiqing Algebraic Geometry 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 $Z_{f,φ}$ 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 $Z_{f,φ}$. 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. |
| title | On the Poles of Real Archimedean Zeta Functions |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.07679 |