Multidimensional residues and discriminants of polynomial mappings
August Tsikh, Krasnoyarsk University, Russia, and Stockholm University, Sweden
Abstract
In this talk we shall discuss the use of multidimensional
residue techniques in the study of multiple Mellin-Barnes integrals. These
integrals represent generalized hypergeometric functions, and their
domains of convergence are determined solely in terms of the coordinates
(arg z1,...,arg zn), where
z is a complex vector
variable in Cn. The projection of the convergence domain onto these argument
coordinates is a convex set which in a natural manner contains several
connected components of the complement of the coamoeba of the singular
hypersurface of the hypergeometric function. The singular hypersurface
itself can be seen as a discriminant (or a principal determinant) of a
polynomial mapping. We will show that the Mellin-Barnes integrals encode
information about the Newton polytope of these discriminants and about the
configuration of their zeros.