26 research outputs found
A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial
The specialized V\'amos polynomial is a hyperbolic polynomial of degree four
in four variables with the property that none of its powers admits a definite
determinantal representation. We will use a heuristical method to prove that
its hyperbolicity cone is a spectrahedron.Comment: Notable easier arguments and minor correction
On the half-plane property and the Tutte group of a matroid
A multivariate polynomial is stable if it is non-vanishing whenever all
variables have positive imaginary parts. A matroid has the weak half-plane
property (WHPP) if there exists a stable polynomial with support equal to the
set of bases of the matroid. If the polynomial can be chosen with all of its
nonzero coefficients equal to one then the matroid has the half-plane property
(HPP). We describe a systematic method that allows us to reduce the WHPP to the
HPP for large families of matroids. This method makes use of the Tutte group of
a matroid. We prove that no projective geometry has the WHPP and that a binary
matroid has the WHPP if and only if it is regular. We also prove that T_8 and
R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.
Obstructions to determinantal representability
There has recently been ample interest in the question of which sets can be
represented by linear matrix inequalities (LMIs). A necessary condition is that
the set is rigidly convex, and it has been conjectured that rigid convexity is
also sufficient. To this end Helton and Vinnikov conjectured that any real zero
polynomial admits a determinantal representation with symmetric matrices. We
disprove this conjecture. By relating the question of finding LMI
representations to the problem of determining whether a polymatroid is
representable over the complex numbers, we find a real zero polynomial such
that no power of it admits a determinantal representation. The proof uses
recent results of Wagner and Wei on matroids with the half-plane property, and
the polymatroids associated to hyperbolic polynomials introduced by Gurvits.Comment: 10 pages. To appear in Advances in Mathematic
Discrete concavity and the half-plane property
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-vanishing properties. This
family contains several of the most 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.Comment: 14 pages. The proof of Theorem 4 is corrected
Matroids arising from electrical networks
This paper introduces Dirichlet matroids, a generalization of graphic
matroids arising from electrical networks. We present four main results. First,
we exhibit a matroid quotient formed by the dual of a network embedded in a
surface with boundary and the dual of the associated Dirichlet matroid. This
generalizes an analogous result for graphic matroids of cellularly embedded
graphs. Second, we characterize the Bergman fans of Dirichlet matroids as
explicit subfans of graphic Bergman fans. In doing so, we generalize the
connection between Bergman fans of complete graphs and phylogenetic trees.
Third, we use the half-plane property of Dirichlet matroids to prove an
interlacing result on the real zeros and poles of the trace of the response
matrix. And fourth, we bound the coefficients of the precoloring polynomial of
a network by the coefficients of the chromatic polynomial of the underlying
graph.Comment: 27 pages, 14 figure