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Discriminants, Symmetrized Graph monomials and Sums of Squares
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 21, no 4, 353-361 p.Article in journal (Refereed) Published
Abstract [en]

In 1878, motivated by the requirements of the invariant the-ory of binary forms, J. J. Sylvester constructed, for every graphwith possible multiple edges but without loops, its symmetrizedgraph monomial, which is a polynomial in the vertex labels ofthe original graph. We pose the question for which graphs thispolynomial is nonnegative or a sum of squares. This problem ismotivated by a recent conjecture of F. Sottile and E. Mukhin onthe discriminant of the derivative of a univariate polynomial andby an interesting example of P. and A. Lax of a graph with fouredges whose symmetrized graph monomial is nonnegative butnot a sum of squares. We present detailed information about sym-metrized graph monomials for graphs with four and six edges,obtained by computer calculations.

Place, publisher, year, edition, pages
2012. Vol. 21, no 4, 353-361 p.
Keyword [en]
polynomial sums of squares, translation-invariant polynomials, graph monomials, sums of squares, discriminants, symmetric polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-84745DOI: 10.1080/10586458.2012.669608ISI: 000313614400003OAI: oai:DiVA.org:su-84745DiVA: diva2:581344
Available from: 2012-12-31 Created: 2012-12-31 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Combinatorial Methods in Complex Analysis
Open this publication in new window or tab >>Combinatorial Methods in Complex Analysis
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts.

Part A: Spectral properties of the Schrödinger equation

This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained.

Part B: Graph monomials and sums of squares

In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares.

Part C: Eigenvalue asymptotics of banded Toeplitz matrices

This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above.

Part D: Stretched Schur polynomials

This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2013. 111 p.
Keyword
combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-88808 (URN)978-91-7447-684-2 (ISBN)
Public defence
2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript

Available from: 2013-05-08 Created: 2013-03-30 Last updated: 2013-05-06Bibliographically approved

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