Tropical aspects of real polynomials and hypergeometric functions
2015 (English)Doctoral thesis, monograph (Other academic)
The present thesis has three main topics: geometry of coamoebas, hypergeometric functions, and geometry of zeros.
First, we study the coamoeba of a Laurent polynomial f in n complex variables. We define a simpler object, which we call the lopsided coamoeba, and associate to the lopsided coamoeba an order map. That is, we give a bijection between the set of connected components of the complement of the closed lopsided coamoeba and a finite set presented as the intersection of an affine lattice and a certain zonotope. Using the order map, we then study the topology of the coamoeba. In particular, we settle a conjecture of M. Passare concerning the number of connected components of the complement of the closed coamoeba in the case when the Newton polytope of f has at most n+2 vertices.
In the second part we study hypergeometric functions in the sense of Gel'fand, Kapranov, and Zelevinsky. We define Euler-Mellin integrals, a family of Euler type hypergeometric integrals associated to a coamoeba. As opposed to previous studies of hypergeometric integrals, the explicit nature of Euler-Mellin integrals allows us to study in detail the dependence of A-hypergeometric functions on the homogeneity parameter of the A-hypergeometric system. Our main result is a complete description of this dependence in the case when A represents a toric projective curve.
In the last chapter we turn to the theory of real univariate polynomials. The famous Descartes' rule of signs gives necessary conditions for a pair (p,n) of integers to represent the number of positive and negative roots of a real polynomial. We characterize which pairs fulfilling Descartes' conditions are realizable up to degree 7, and we provide restrictions valid in arbitrary degree.
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, 2015. , 124 p.
, Stockholm dissertations in mathematics, 55
Amoeba, Tropical Geometry, Hypergeometric function, Geometry of zeros, Discriminant
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-116358ISBN: 978-91-7649-173-7OAI: oai:DiVA.org:su-116358DiVA: diva2:807588
2015-06-04, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Mikhalkin, Grigory, Professor
Shapiro, Boris, Professor