Change search
ReferencesLink to record
Permanent link

Direct link
Tropical aspects of real polynomials and hypergeometric functions
Stockholm University, Faculty of Science, Department of Mathematics.
2015 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

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
Keyword [en]
Amoeba, Tropical Geometry, Hypergeometric function, Geometry of zeros, Discriminant
National Category
Research subject
URN: urn:nbn:se:su:diva-116358ISBN: 978-91-7649-173-7OAI: diva2:807588
Public defence
2015-06-04, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Available from: 2015-05-12 Created: 2015-04-20 Last updated: 2015-06-16Bibliographically approved

Open Access in DiVA

fulltext(3067 kB)316 downloads
File information
File name FULLTEXT01.pdfFile size 3067 kBChecksum SHA-512
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Forsgård, Jens
By organisation
Department of Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 316 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 820 hits
ReferencesLink to record
Permanent link

Direct link