6,155 research outputs found
The symplectic and twistor geometry of the general isomonodromic deformation problem
Hitchin's twistor treatment of Schlesinger's equations is extended to the
general isomonodromic deformation problem. It is shown that a generic linear
system of ordinary differential equations with gauge group SL(n,C) on a Riemann
surface X can be obtained by embedding X in a twistor space Z on which sl(n,C)
acts. When a certain obstruction vanishes, the isomonodromic deformations are
given by deforming X in Z. This is related to a description of the deformations
in terms of Hamiltonian flows on a symplectic manifold constructed from affine
orbits in the dual Lie algebra of a loop group.Comment: 35 pages, LATE
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
On Generalized Gauge-Fixing in the Field-Antifield Formalism
We consider the problem of covariant gauge-fixing in the most general setting
of the field-antifield formalism, where the action W and the gauge-fixing part
X enter symmetrically and both satisfy the Quantum Master Equation. Analogous
to the gauge-generating algebra of the action W, we analyze the possibility of
having a reducible gauge-fixing algebra of X. We treat a reducible gauge-fixing
algebra of the so-called first-stage in full detail and generalize to arbitrary
stages. The associated "square root" measure contributions are worked out from
first principles, with or without the presence of antisymplectic second-class
constraints. Finally, we consider an W-X alternating multi-level
generalization.Comment: 49 pages, LaTeX. v2: Minor changes + 1 more reference. v3,v4,v5:
Corrected typos. v5: Version published in Nuclear Physics B. v6,v7:
Correction to the published version added next to the Acknowledgemen
Improved method for finding optimal formulae for bilinear maps in a finite field
In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework
to exhaustively search for optimal formulae for evaluating bilinear maps, such
as Strassen or Karatsuba formulae. The main contribution of this work is a new
criterion to aggressively prune useless branches in the exhaustive search, thus
leading to the computation of new optimal formulae, in particular for the short
product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we
are able to prove that there is essentially only one optimal decomposition of
the product of 3 x 2 by 2 x 3 matrices up to the action of some group of
automorphisms
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