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Spectrahedrality of hyperbolicity cones of multivariate matching polynomials
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2018 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Refereed) In press
Abstract [en]

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

Place, publisher, year, edition, pages
Springer, 2018.
Keywords [en]
matching polynomial, independence polynomial, generalized Lax conjecture, spectrahedral representation
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-250763DOI: 10.1007/s10801-018-0848-9OAI: oai:DiVA.org:kth-250763DiVA, id: diva2:1313629
Note

QC 20190510

Available from: 2019-05-05 Created: 2019-05-05 Last updated: 2019-10-11Bibliographically approved
In thesis
1. Combinatorics and zeros of multivariate polynomials
Open this publication in new window or tab >>Combinatorics and zeros of multivariate polynomials
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 42
Series
TRITA-SCI-FOU ; 2019:33
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-251303 (URN)978-91-7873-210-4 (ISBN)
Public defence
2019-05-24, D3, Lindstedsvägen 5, Stockholm, 14:00 (English)
Opponent
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Note

QC 20190510

Available from: 2019-05-10 Created: 2019-05-09 Last updated: 2019-05-10Bibliographically approved

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Amini, Nima

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