On the Poles of Real Archimedean Zeta Functions

2025/12/08 by Chen, Zhikuang, Zuo, Huaiqing
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.07679

Abstract

This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial f ∈ 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 Re s > -1. This explicit expression yields a necessary and sufficient condition for a root s ∈ (-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.

Citations

• Polar loci of multivariable archimedean zeta functions
• Archimedean zeta functions, singularities, and Hodge theory
• Variation of Archimedean Zeta Function and n/d-Conjecture for Generic Multiplicities
• Monodromy conjecture for semi-quasihomogeneous hypersurfaces
• Zero loci of Bernstein-Sato ideals

Related

• Archimedean zeta functions, singularities, and Hodge theory
• Poles of the complex zeta function of a plane curve
• Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci
• Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities
• Bernstein-Sato polynomials of semi-weighted-homogeneous polynomials of nearly Brieskorn-Pham type