Mellin Transforms of Multivariate Rational Functions |
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Authors: | Lisa Nilsson Mikael Passare |
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Institution: | 1. Mathematical Sciences, Chalmers University of Technology and Gothenburg University, 412 96, Gothenburg, Sweden 2. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden
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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 $\mathcal{A}'_{f}:=\mathrm{Arg}(Z_{f})$ , 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 $\mathcal{A}'_{f}$ 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. |
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