Euler products associated to multivariate rational functions: maximal domain of meromorphy, zeros and poles

Chadozeau A.; Velásquez Castañón O. · Dissertationes Mathematicae (2026) · DOI: 10.4064/dm240206-23-2

Resumen

We determine the maximal domain of meromorphy of any Euler product (Formula presented), where h is the quotient of two polynomials in k variables, with integer coefficients and with constant term equal to 1. More precisely, we define a domain (Formula presented) , on which f admits a meromorphic extension, and such that for any z in ∂Γ, there is no neighborhood of z on which f admits a meromorphic extension. This maximal domain Γ is described as (Formula presented), where (Formula presented) is a rational cone, computed from the set of exponents of the two polynomials defining h. We also describe the divisor of f over Γ, which comes from the local factors (Formula presented), and from the zeta factors (Formula presented), where ζ denotes the Riemann ζ-function, corresponding to terms in the expansion of h as a formal infinite product (Formula presented). We focus our study on the hyperplanes in the divisor, allowing us to use tools developed for the single variable case. We complete our study by giving a geometric and arithmetic description of the set of exponents occurring in the infinite product expansion of h, and by showing a new result on the geometric nature of the set of singular points of a holomorphic function defined over a tubular domain. © Copyright by Instytut Matematyczny PAN, Warszawa 2026

Instituciones: Universidad Nacional de Ingeniería