×

zbMATH — the first resource for mathematics

Mellin transforms of multivariate rational functions. (English) Zbl 1271.44001
If \(h\) is a locally integrable function on the positive real axis, the Mellin transform is defined by the formula \[ M_h(s)= \int^\infty_0 h(z)\,z^s{dz\over z}, \] where \(s= \sigma+ it\).
This transform is closely related to the Fourier-Laplace transform.
In the paper, the authors consider Mellin transforms of rational functions \(h= g/f\), where \(g\) and \(f\) are polynomials, among them in several variables.
It is proved that the polar set of such Mellin transforms consists of finitely many families of parallel hyperplanes, with all planes in each family being integral translates of a specific facial hyperplane of the Newton polytope of the denominator \(f\).
The Mellin transform is related to the so-called coamoeba and the theory of \(A\)-hypergeometric functions.

MSC:
44A15 Special integral transforms (Legendre, Hilbert, etc.)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Andersson, M., Passare, M., Sigurdsson, R.: Complex Convexity and Analytic Functionals. Progress in Mathematics, vol. 225. Birkhäuser Verlag, Basel (2004). xii+160 pp. · Zbl 1057.32001
[2] Antipova, I.: Inversions of multidimensional Mellin transforms and solutions of algebraic equations. Sb. Math. 198, 447–463 (2007) · Zbl 1142.44006
[3] Beukers, F.: Algebraic A-hypergeometric functions. Invent. Math. 180, 589–610 (2010) · Zbl 1251.33011
[4] Bochner, S.: A theorem on analytic continuation of functions in several variables. Ann. Math. 39, 1–19 (1938) · JFM 64.0321.02
[5] Ermolaeva, T., Tsikh, A.: Integration of rational functions over \(\mathbb{R}\) n by means of toric compactifications and multidimensional residues. Sb. Math. 187(9), 1301–1318 (1996) · Zbl 0876.32002
[6] Forsberg, M., Passare, M., Tsikh, A.: Laurent determinants and arrangements of hyperplane amoebas. Adv. Math. 151, 45–70 (2000) · Zbl 1002.32018
[7] Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants and Multidimensional Determinants. Modern Birkhäuser Classics (2008). Boston, x+523 pp. · Zbl 1138.14001
[8] Gelfand, I., Kapranov, M., Zelevinsky, A.: Hypergeometric functions and toric varieties. Funct. Anal. Appl. 23, 94–106 (1989) · Zbl 0721.33006
[9] Gelfand, I., Kapranov, M., Zelevinsky, A.: Generalized Euler integrals and A-hypergeometric functions. Adv. Math. 84, 255–271 (1990) · Zbl 0741.33011
[10] Hörmander, L.: The Analysis of Linear Partial Differential Operators 1. Classics in Mathematics. Springer, Berlin, (2003). x+440 pp. · Zbl 1028.35001
[11] Hörmander, L.: Notions of convexity. Modern Birkhäuser Classics (2007). Boston, viii+414 pp.
[12] Johansson, P.: Coamoebas. Licentiate thesis, Department of Mathematics, Stockholm University (2010)
[13] Nilsson, L.: Amoebas, discriminants, and hypergeometric functions. Doctoral thesis, Department of Mathematics, Stockholm University (2009)
[14] Nisse, M., Sottile, F.: The phase limit set of a variety. Manuscript, 2011 · Zbl 1277.14048
[15] Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations. Algorithms and Computation in Mathematics, vol. 6. Springer, Berlin (2000). viii+254 pp. · Zbl 0946.13021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.