Mellin transforms of multivariate rational functions

2010/10/25 by Lisa M. Nilsson, Mikael Passare, Nilsson, Lisa +1 · 6 citations
Engineering · Mathematics · #32 #51 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1010.5060

openalex publication_date 2010/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper deals with Mellin transforms of rational functions g/f in several variables. We prove that the polar set of such a Mellin transform consists of finitely many families of parallel hyperplanes, with all planes in each such family being integral translates of a specific facial hyperplane of the Newton polytope of the denominator f. The Mellin transform is naturally related to the so called coamoeba A'_f:=Arg (Z_f), where Z_f is the zero locus of f and Arg denotes the mapping that takes each coordinate to its argument. In fact, each connected component of the complement of the coamoeba A'_f gives rise to a different Mellin transform. The dependence of the Mellin transform on the coefficients of f, and the relation to the theory of A-hypergeometric functions is also discussed in the paper.

Cited by

• Characterizing Cohen-Macaulay One-Loop Feynman Integrals
• Fano and Reflexive Polytopes from Feynman Integrals
• Expansion by Regions Derivation from the Mellin Transform
• Tropical regions of near mass-shell pentabox
• Approximating amoebas and coamoebas by sums of squares
• Off-shell form factor: factorization is violated

Related

• Euler--Mellin integrals and A-hypergeometric functions
• Mellin transforms with only critical zeros: Legendre functions
• Mellin transforms of p-adic Whittaker functions
• Mellin transforms of power-constructible functions
• Special functions and the Mellin transforms of Laguerre and Hermite functions