38,846 research outputs found
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
Haantjes Algebras of Classical Integrable Systems
A tensorial approach to the theory of classical Hamiltonian integrable
systems is proposed, based on the geometry of Haantjes tensors. We introduce
the class of symplectic-Haantjes manifolds (or manifolds),
as the natural setting where the notion of integrability can be formulated. We
prove that the existence of suitable Haantjes algebras of (1,1) tensor fields
with vanishing Haantjes torsion is a necessary and sufficient condition for a
Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show
that new integrable models arise from the Haantjes geometry. Finally, we
present an application of our approach to the study of the Post-Winternitz
system and of a stationary flow of the KdV hierarchy.Comment: 31 page
The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem
We review the Calogero-Fran\c{c}oise integrable system, which is a
generalization of the Camassa-Holm system. We express solutions as (twisted)
Higgs bundles, in the sense of Hitchin, over the projective line. We use this
point of view to (a) establish a general answer to the question of
linearization of isospectral flow and (b) demonstrate, in the case of two
particles, the dynamical meaning of the theta divisor of the spectral curve in
terms of mechanical collisions. Lastly, we outline the solution to the inverse
problem for CF flows using Stieltjes' continued fractions.Comment: 22 pages, 2 figure
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
Vector bundles and Lax equations on algebraic curves
The Hamiltonian theory of zero-curvature equations with spectral parameter on
an arbitrary compact Riemann surface is constructed. It is shown that the
equations can be seen as commuting flows of an infinite-dimensional field
generalization of the Hitchin system. The field analog of the elliptic
Calogero-Moser system is proposed. An explicit parameterization of Hitchin
system based on the Tyurin parameters for stable holomorphic vector bundles on
algebraic curves is obtained.Comment: Latex, 42page
Information measures and classicality in quantum mechanics
We study information measures in quantu mechanics, with particular emphasis
on providing a quantification of the notions of classicality and
predictability. Our primary tool is the Shannon - Wehrl entropy I. We give a
precise criterion for phase space classicality and argue that in view of this
a) I provides a measure of the degree of deviation from classicality for closed
system b) I - S (S the von Neumann entropy) plays the same role in open systems
We examine particular examples in non-relativistic quantum mechanics. Finally,
(this being one of our main motivations) we comment on field classicalisation
on early universe cosmology.Comment: 35 pages, LATE
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