Polynomials with the half-plane property and the support theorems

Choe, Youngbin

doi:10.1016/j.jctb.2004.12.001

Cited by 9 publications

(14 citation statements)

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“…The half-plane property. Let us recall that a polynomial P with complex coefficients is said to have the half-plane property [42,41,142,28,27,145,30,144] if either P ≡ 0 or else P (x 1 , . .…”

Section: Some Further Remarksmentioning

confidence: 99%

Complete monotonicity for inverse powers of some combinatorially defined polynomials

Scott

Sokal

2014

Acta Math.

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We prove the complete monotonicity on (0, ∞) n for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab initio methods for proving that P −β is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of P −β for some β > 0 can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P , and is also related to the Rayleigh property for matroids.

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Section: Some Further Remarksmentioning

confidence: 99%

Complete monotonicity for inverse powers of some combinatorially defined polynomials

Scott

Sokal

2014

Acta Math.

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“…If H is the upper half-plane we say that f is stable 1 , and if H is the right half-plane that f is Hurwitz stable. Multivariate polynomials with the half-plane property appear (sometimes hidden) in many different areas such as statistical mechanics [14,20,25], complex analysis [16,21], differential equations [1,11], engineering [9,19], optimization [13] and combinatorics [5,6,7,13,14,31,32]. Recently a striking correspondence between polynomials with the half-plane property and matroids was found [6].…”

Section: Introductionmentioning

confidence: 99%

“…property and matroid theory has since then been continued in a series of papers [5,7,13,31,32] where several interesting open questions have been raised. In this paper we answer some of these open questions and pose others.…”

Section: Introductionmentioning

confidence: 99%

Polynomials with the half-plane property and matroid theory

Brändén

2007

Advances in Mathematics

130

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A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * We characterize multivariate multi-affine polynomial with real coefficients that have the half-plane property (with respect to the upper half-plane) in terms of inequalities. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. * We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat

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“…Recent (mostly algebraic) research (Brändén 2007;Choe 2005) for properly understanding the relation of the last three classes revealed further connections with the theory of jump systems Lovász 1997;Recski and Szabó 2006). …”

Section: Examplesmentioning

confidence: 97%