Multidimensional residues and discriminants of polynomial mappings
August Tsikh, Krasnoyarsk University, Russia, and Stockholm University, Sweden
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.