16,976 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
Classification of unital simple Leavitt path algebras of infinite graphs
We prove that if E and F are graphs with a finite number of vertices and an
infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple
Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only
if K_0^\textnormal{alg} (L_K(E)) \cong K_0^\textnormal{alg} (L_K(F)) and the
graphs and have the same number of singular vertices, and moreover, in
this case one may transform the graph E into the graph F using basic moves that
preserve the Morita equivalence class of the associated Leavitt path algebra.
We also show that when K is a field with no free quotients, the condition that
E and F have the same number of singular vertices may be replaced by
K_1^\textnormal{alg} (L_K(E)) \cong K_1^\textnormal{alg} (L_K(F)), and we
produce examples showing this cannot be done in general. We describe how we can
combine our results with a classification result of Abrams, Louly, Pardo, and
Smith to get a nearly complete classification of unital simple Leavitt path
algebras - the only missing part is determining whether the "sign of the
determinant condition" is necessary in the finite graph case. We also consider
the Cuntz splice move on a graph and its effect on the associated Leavitt path
algebra.Comment: Version IV Comments: We have made some substantial revisions, which
include extending our classification results to Leavitt path algebras over
arbitrary fields. This is the version that will be published. Version III
Comments: Some typos and errors corrected. New section (Section 10) has been
added. Some references added. Version II Comments: Some typos correcte
QCD at non-zero density and canonical partition functions with Wilson fermions
We present a reduction method for Wilson Dirac fermions with non-zero
chemical potential which generates a dimensionally reduced fermion matrix. The
size of the reduced fermion matrix is independent of the temporal lattice
extent and the dependence on the chemical potential is factored out. As a
consequence the reduced matrix allows a simple evaluation of the Wilson fermion
determinant for any value of the chemical potential and hence the exact
projection to the canonical partition functions.Comment: 22 pages, 11 figures, 1 table; references added, figure size reduce
Quaternary quadratic lattices over number fields
We relate proper isometry classes of maximal lattices in a totally definite
quaternary quadratic space (V,q) with trivial discriminant to certain
equivalence classes of ideals in the quaternion algebra representing the
Clifford invariant of (V,q). This yields a good algorithm to enumerate a system
of representatives of proper isometry classes of lattices in genera of maximal
lattices in (V,q)
Black Box Galois Representations
We develop methods to study -dimensional -adic Galois representations
of the absolute Galois group of a number field , unramified outside a
known finite set of primes of , which are presented as Black Box
representations, where we only have access to the characteristic polynomials of
Frobenius automorphisms at a finite set of primes. Using suitable finite test
sets of primes, depending only on and , we show how to determine the
determinant , whether or not is residually reducible, and
further information about the size of the isogeny graph of whose
vertices are homothety classes of stable lattices. The methods are illustrated
with examples for , and for imaginary quadratic, being
the representation attached to a Bianchi modular form.
These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees'
report
Evans function and Fredholm determinants
We explore the relationship between the Evans function, transmission
coefficient and Fredholm determinant for systems of first order linear
differential operators on the real line. The applications we have in mind
include linear stability problems associated with travelling wave solutions to
nonlinear partial differential equations, for example reaction-diffusion or
solitary wave equations. The Evans function and transmission coefficient, which
are both finite determinants, are natural tools for both analytic and numerical
determination of eigenvalues of such linear operators. However, inverting the
eigenvalue problem by the free state operator generates a natural linear
integral eigenvalue problem whose solvability is determined through the
corresponding infinite Fredholm determinant. The relationship between all three
determinants has received a lot of recent attention. We focus on the case when
the underlying Fredholm operator is a trace class perturbation of the identity.
Our new results include: (i) clarification of the sense in which the Evans
function and transmission coefficient are equivalent; and (ii) proof of the
equivalence of the transmission coefficient and Fredholm determinant, in
particular in the case of distinct far fields.Comment: 26 page
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