1,706 research outputs found
Computing in Jacobians of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over
finite fields, and with their Jacobians, using the algorithmic representation
of projective curves developed by Khuri-Makdisi. We show that many desirable
operations can be done efficiently in this setting: decomposing divisors into
prime divisors; computing pull-backs and push-forwards of divisors under finite
morphisms, and hence Picard and Albanese maps on Jacobians; generating
uniformly random divisors and points on Jacobians; computing Frobenius maps and
Kummer maps; and finding a basis for the -torsion of the Picard group, where
is a prime number different from the characteristic of the base field.Comment: 42 page
Matrix Pencils and Entanglement Classification
In this paper, we study pure state entanglement in systems of dimension
. Two states are considered equivalent if they can be
reversibly converted from one to the other with a nonzero probability using
only local quantum resources and classical communication (SLOCC). We introduce
a connection between entanglement manipulations in these systems and the
well-studied theory of matrix pencils. All previous attempts to study general
SLOCC equivalence in such systems have relied on somewhat contrived techniques
which fail to reveal the elegant structure of the problem that can be seen from
the matrix pencil approach. Based on this method, we report the first
polynomial-time algorithm for deciding when two states
are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC
equivalence classes in systems, we also determine the
hierarchy between these classes
Computing Minimal Polynomials of Matrices
We present and analyse a Monte-Carlo algorithm to compute the minimal
polynomial of an matrix over a finite field that requires
field operations and O(n) random vectors, and is well suited for successful
practical implementation. The algorithm, and its complexity analysis, use
standard algorithms for polynomial and matrix operations. We compare features
of the algorithm with several other algorithms in the literature. In addition
we present a deterministic verification procedure which is similarly efficient
in most cases but has a worst-case complexity of . Finally, we report
the results of practical experiments with an implementation of our algorithms
in comparison with the current algorithms in the {\sf GAP} library
A Survey on Fixed Divisors
In this article, we compile the work done by various mathematicians on the
topic of the fixed divisor of a polynomial. This article explains most of the
results concisely and is intended to be an exhaustive survey. We present the
results on fixed divisors in various algebraic settings as well as the
applications of fixed divisors to various algebraic and number theoretic
problems. The work is presented in an orderly fashion so as to start from the
simplest case of progressively leading up to the case of Dedekind
domains. We also ask a few open questions according to their context, which may
give impetus to the reader to work further in this direction. We describe
various bounds for fixed divisors as well as the connection of fixed divisors
with different notions in the ring of integer-valued polynomials. Finally, we
suggest how the generalization of the ring of integer-valued polynomials in the
case of the ring of matrices over (or Dedekind domain) could
lead to the generalization of fixed divisors in that setting.Comment: Accepted for publication in Confluentes Mathematic
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