Determinantal representations and Bézoutians

Kummer, Mario

doi:10.1007/s00209-016-1715-9

Cited by 19 publications

(16 citation statements)

References 28 publications

(23 reference statements)

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“…• Conjecture 2.3 is true for elementary symmetric polynomials, see [6], • Weaker versions of Conjecture 2.3 are true for smooth hyperbolic polynomials, see [25,28]. • Stronger algebraic versions of Conjecture 2.3 are false, see [5].…”

Section: Hyperbolic and Stable Polynomialsmentioning

confidence: 99%

Non-representable hyperbolic matroids

Amini

Brändén

2018

Advances in Mathematics

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The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vámos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids.We prove that the Non-Pappus and Non-Desargues matroids are nonrepresentable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vámos matroid and the generalized Vámos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre-Turán inequality and Jensen's inequality. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

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Section: Hyperbolic and Stable Polynomialsmentioning

confidence: 99%

Non-representable hyperbolic matroids

Amini

Brändén

2018

Advances in Mathematics

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“…In this section we prove a Positivstellensatz for matrices over a ring A similar to Krivine's Positivstellensatz. Note that this is an ungraded version of the Positivstellensatz proved and used in [Kum13] that holds over the graded ring of real polynomials. The proof is also very similar.…”

Section: A Positivstellensatzmentioning

confidence: 99%

Eigenvalues of symmetric matrices over integral domains

Kummer

2016

Journal of Algebra

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Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element. The latter result settles a questions raised by Bass, Estes and Guralnick.Comment: Serious error in previous versio

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“…Since our generalized Dixon process does not require the interlacer to be contact, it is possible that a spectrahedral description of the hyperbolicity cone could be constructed in a similar way, but this is currently purely speculative. In §2.1 we point out how our construction is related to sum-ofsquares decompositions of Bézout matrices and the construction in [9].…”

Section: Introductionmentioning

confidence: 99%

Spectrahedral representations of plane hyperbolic curves

2019

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We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.It coincides with the hyperbolicity cone C(f, e) of f = det M in direction e, that is the closure of the connected component of {a ∈ R 3 : f (a) = 0} containing e. This is a convex cone in R 3 , whose image in P 2 is the region enclosed by the convex innermost oval of the curve (see Figure 1). A triple of real symmetric matrices A, B, C is a spectrahedral representation of C(f, e) if M = xA + yB + zC satisfies C(f, e) = S(M ).

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Determinantal representations and Bézoutians

Cited by 19 publications

References 28 publications

Non-representable hyperbolic matroids

Non-representable hyperbolic matroids

Eigenvalues of symmetric matrices over integral domains

Spectrahedral representations of plane hyperbolic curves

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