217 research outputs found
On semiring complexity of Schur polynomials
Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables is fixed
Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
Integrable probability has emerged as an active area of research at the
interface of probability/mathematical physics/statistical mechanics on the one
hand, and representation theory/integrable systems on the other. Informally,
integrable probabilistic systems have two properties: 1) It is possible to
write down concise and exact formulas for expectations of a variety of
interesting observables (or functions) of the system. 2) Asymptotics of the
system and associated exact formulas provide access to exact descriptions of
the properties and statistics of large universality classes and universal
scaling limits for disordered systems. We focus here on examples of integrable
probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality
class and explain how their integrability stems from connections with symmetric
function theory and quantum integrable systems.Comment: Proceedings of the ICM, 31 pages, 10 figure
Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn\u27t
Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Stanley, 1999], in Schubert calculus [Ledoux and Malham, 2010] as well as in Enumerative combinatorics [Gasharov, 1996; Stanley, 1984; Stanley, 1999]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [Hallgren et al., 2000; Ryan O\u27Donnell and John Wright, 2015] and Geometric complexity theory [Ikenmeyer and Panova, 2017].
However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas.
As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest
Tropical secant graphs of monomial curves
The first secant variety of a projective monomial curve is a threefold with
an action by a one-dimensional torus. Its tropicalization is a
three-dimensional fan with a one-dimensional lineality space, so the tropical
threefold is represented by a balanced graph. Our main result is an explicit
construction of that graph. As a consequence, we obtain algorithms to
effectively compute the multidegree and Chow polytope of an arbitrary
projective monomial curve. This generalizes an earlier degree formula due to
Ranestad. The combinatorics underlying our construction is rather delicate, and
it is based on a refinement of the theory of geometric tropicalization due to
Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular,
old Sections 4 and 5 are merged into a single section. Also, added Figure 3
and discussed Chow polytopes of rational normal curves in Section
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
Recommended from our members
Reelle Algebraische Geometrie
This workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
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