53 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
A converse to the Grace--Walsh--Szeg\H{o} theorem
We prove that the symmetrizer of a permutation group preserves stability of a
polynomial if and only if the group is orbit homogeneous. A consequence is that
the hypothesis of permutation invariance in the Grace-Walsh-Szeg\H{o}
Coincidence Theorem cannot be relaxed. In the process we obtain a new
characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle,
and prove that the class of such polynomials can be endowed with a natural
multiplication.Comment: 7 page
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
Hyperbolic Polynomials and Generalized Clifford Algebras
We consider the problem of realizing hyperbolicity cones as spectrahedra,
i.e. as linear slices of cones of positive semidefinite matrices. The
generalized Lax conjecture states that this is always possible. We use
generalized Clifford algebras for a new approach to the problem. Our main
result is that if -1 is not a sum of hermitian squares in the Clifford algebra
of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our
result also has computational applications, since this sufficient condition can
be checked with a single semidefinite program
A Reciprocity Theorem for Monomer-Dimer Coverings
The problem of counting monomer-dimer coverings of a lattice is a
longstanding problem in statistical mechanics. It has only been exactly solved
for the special case of dimer coverings in two dimensions. In earlier work,
Stanley proved a reciprocity principle governing the number of dimer
coverings of an by rectangular grid (also known as perfect matchings),
where is fixed and is allowed to vary. As reinterpreted by Propp,
Stanley's result concerns the unique way of extending to so
that the resulting bi-infinite sequence, for , satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that is always an integer satisfying the relation where unless 2(mod 4) and
is odd, in which case . Furthermore, Propp's method is
applicable to higher-dimensional cases. This paper discusses similar
investigations of the numbers , of monomer-dimer coverings, or
equivalently (not necessarily perfect) matchings of an by rectangular
grid. We show that for each fixed there is a unique way of extending
to so that the resulting bi-infinite sequence, for , satisfies a linear recurrence relation with constant coefficients. We
show that , a priori a rational number, is always an integer, using a
generalization of the combinatorial model offered by Propp. Lastly, we give a
new statement of reciprocity in terms of multivariate generating functions from
which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes
Correlation bounds for fields and matroids
Let be a finite connected graph, and let be a spanning tree of
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events and are negatively
correlated for any distinct edges and . What can be said for such
events when the underlying matroid is not necessarily graphic? We use Hodge
theory for matroids to bound the correlation between the events ,
where is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of -element independent sets of a matroid
forms an ultra-log-concave sequence in .Comment: 16 pages. Supersedes arXiv:1804.0307
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