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Discrete concavity and the half-plane property
Department of Mathematics, Stockholm University.ORCID iD: 0000-0003-1055-1474
2010 (English)In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 24, no 3, 921-933 p.Article in journal (Refereed) Published
Abstract [en]

Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed non-anishing properties. This family contains several of the most well studied M-concave functions in the literature. In the language of tropical geometry, we study the tropicalization of the space of polynomials with the half-plane property and show that it is strictly contained in the space of M-concave functions. We also provide a short proof of Speyer's "hive theorem" which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices.

Place, publisher, year, edition, pages
2010. Vol. 24, no 3, 921-933 p.
Keyword [en]
M-convex, jump system, matroid, half-plane property, tropicalization, Puiseux series, Tarski's principle, hive, Horn's conjecture
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-78730DOI: 10.1137/090758738ISI: 000282291600014OAI: oai:DiVA.org:kth-78730DiVA: diva2:492816
Note
QC 20120221Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2017-12-08Bibliographically approved

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Brändén, Petter

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