1,826 research outputs found
Uniform determinantal representations
The problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
Orbit measures, random matrix theory and interlaced determinantal processes
A connection between representation of compact groups and some invariant
ensembles of Hermitian matrices is described. We focus on two types of
invariant ensembles which extend the Gaussian and the Laguerre Unitary
ensembles. We study them using projections and convolutions of invariant
probability measures on adjoint orbits of a compact Lie group. These measures
are described by semiclassical approximation involving tensor and restriction
mulltiplicities. We show that a large class of them are determinantal
Determinantal Processes and Independence
We give a probabilistic introduction to determinantal and permanental point
processes. Determinantal processes arise in physics (fermions, eigenvalues of
random matrices) and in combinatorics (nonintersecting paths, random spanning
trees). They have the striking property that the number of points in a region
is a sum of independent Bernoulli random variables, with parameters which
are eigenvalues of the relevant operator on . Moreover, any
determinantal process can be represented as a mixture of determinantal
projection processes. We give a simple explanation for these known facts, and
establish analogous representations for permanental processes, with geometric
variables replacing the Bernoulli variables. These representations lead to
simple proofs of existence criteria and central limit theorems, and unify known
results on the distribution of absolute values in certain processes with
radially symmetric distributions.Comment: Published at http://dx.doi.org/10.1214/154957806000000078 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
Determinantal representations of semi-hyperbolic polynomials
We prove a generalization of the Hermitian version of the Helton-Vinnikov
determinantal representation of hyperbolic polynomials to the class of
semi-hyperbolic polynomials, a strictly larger class, as shown by an example.
We also prove that certain hyperbolic polynomials affine in two out of four
variables divide a determinantal polynomial. The proofs are based on work
related to polynomials with no zeros on the bidisk and tridisk.Comment: 14 pages, revisio
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