48,525 research outputs found
Permutation-invariant qudit codes from polynomials
A permutation-invariant quantum code on qudits is any subspace stabilized
by the matrix representation of the symmetric group as permutation
matrices that permute the underlying subsystems. When each subsystem is a
complex Euclidean space of dimension , any permutation-invariant code
is a subspace of the symmetric subspace of We give an
algebraic construction of new families of of -dimensional
permutation-invariant codes on at least qudits that can also
correct errors for . The construction of our codes relies on a
real polynomial with multiple roots at the roots of unity, and a sequence of
real polynomials that satisfy some combinatorial constraints. When , we prove constructively that an uncountable number of such
codes exist.Comment: 14 pages. Minor corrections made, to appear in Linear Algebra and its
Application
Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”
This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only “by values”. This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points
Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena
We introduce a family of polynomials, which arise in three distinct ways: in
the large expansion of a matrix integral, as a weighted enumeration of
factorisations of permutations, and via the topological recursion. More
explicitly, we interpret the complex Grassmannian as the
space of idempotent Hermitian matrices of rank and develop a
Weingarten calculus to integrate products of matrix elements over it. In the
regime of large and fixed ratio , such integrals have
expansions whose coefficients count factorisations of permutations into
monotone sequences of transpositions, with each sequence weighted by a monomial
in . This gives rise to the desired polynomials, which
specialise to the monotone Hurwitz numbers when .
These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join
recursion, a one-point recursion, and the topological recursion. Furthermore,
we conjecture on the basis of overwhelming empirical evidence that the deformed
monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy
remarkable interlacing phenomena.
An outcome of our work is the viewpoint that the topological recursion can be
used to "topologise" sequences of polynomials, and we claim that the resulting
families of polynomials may possess interesting properties. As a further case
study, we consider a weighted enumeration of dessins d'enfant and conjecture
that the resulting polynomials are also real-rooted and satisfy analogous
interlacing properties.Comment: 30 pages, 3 figures. Comments welcome
Computing a Hurwitz factorization of a polynomial
AbstractA polynomial is called a Hurwitz polynomial (sometimes, when the coefficients are real, a stable polynomial) if all its roots have real part strictly less than zero. In this paper we present a numerical method for computing the coefficients of the Hurwitz factor f(z) of a polynomial p(z). It is based on a polynomial description of the classical LR algorithm for solving the matrix eigenvalue problem. Similarly with the matrix iteration, it turns out that the proposed scheme has a global linear convergence and, moreover, the convergence rate can be improved by considering the technique of shifting. Our numerical experiments, performed with several test polynomials, indicate that the algorithm has good stability properties since the computed approximation errors are generally in accordance with the estimated condition numbers of the desired factors
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