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Coefficients and zeros of mixed characteristicpolynomials
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Koefficienter och nollställen hos blandadekaraktäristiska polynom (Swedish)
Abstract [en]

The mixed characteristic polynomial (MCP) was introduced in the papersof Marcus, Spielman and Srivastava from 2013 on Ramanujan graphs and the Kadison-Singerconjecture. Several known results and open problems can be formulated in termsof MCPs. The proofs of Marcus, Spielman and Srivastava involve bounding theroots of certain MCPs. Gurvits’ generalization of van der Waerden’s permanentconjecture bounds the constant term of MCPs using the capacity of an underlyingpolynomial.This thesis surveys selected results for MCPs. A counterexample to theHolens-Ðoković conjecture, due to Wanless, is discussed in the context of MCPs.It is used to show how a sequence of MCP coefficients is not monotoneand how the roots of associated Laguerre polynomials do not always majorizethose of other MCPs. Finally, we prove an analogue of the root bound in theproof of the Kadison-Singer conjecture. It applies to product polynomials ofdoubly stochastic matrices through classical results in graph theory due toGodsil, Mohar, Heilmann and Lieb.

Abstract [sv]

Blandade karaktäristiska polynom (MCP) introducerades i Marcus, Spielman och Srivastavas artiklar från 2013 om Ramanujan-grafer och Kadison-Singers förmodan. Såväl kända resultat som öppna problem kan formuleras i termer av koefficienter och rötter hos MCP:er. Bevisen av Marcus, Spielman och Srivastava handlar alla i någon mån om att begränsa rötter till vissa MCP:er. Gurvits generalisering av van der Waerdens permanentförmodan begränsar den konstanta termen hos MCP:er med hjälp av kapaciteten hos ett underliggande polynom.Denna uppsats sammanfattar utvalda resultat om blandade karaktäristiska polynom. Wanless motexempel till en förmodan av Holens och Ðoković diskuteras i detta sammanhang. Det leder till motexempel till en förmodad monotonicitet hos en följd av koefficienter till MCP:er samt till att rötterna hos associerade Laguerrepolynom skulle majorisera de hos andra MCP:er. Slutligen bevisar vi en begränsning av rötterna hos MCP:er till produktpolynom av dubbelt stokastiska matriser motsvarande den gräns som uppstår i beviset av Kadison-Singers förmodan. Beviset bygger på klassiska resultat ur grafteorin av Godsil, Mohar, Heilmann och Lieb.

Place, publisher, year, edition, pages
2017.
Series
TRITA-MAT-E, 2017:30
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-208306OAI: oai:DiVA.org:kth-208306DiVA: diva2:1106332
Subject / course
Mathematics
Educational program
Master of Science - Mathematics
Supervisors
Examiners
Available from: 2017-06-07 Created: 2017-06-07 Last updated: 2017-06-07Bibliographically approved

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