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Tropical aspects of real polynomials and hypergeometric functionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, 2015. , 124 p.
##### Series

, Stockholm dissertations in mathematics, 55
##### Keyword [en]

Amoeba, Tropical Geometry, Hypergeometric function, Geometry of zeros, Discriminant
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-116358ISBN: 978-91-7649-173-7OAI: oai:DiVA.org:su-116358DiVA: diva2:807588
##### Public defence

2015-06-04, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Opponent

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#####

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Available from: 2015-05-12 Created: 2015-04-20 Last updated: 2015-06-16Bibliographically approved

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.

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