178 research outputs found
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
Resolving zero-divisors using Hensel lifting
Algorithms which compute modulo triangular sets must respect the presence of
zero-divisors. We present Hensel lifting as a tool for dealing with them. We
give an application: a modular algorithm for computing GCDs of univariate
polynomials with coefficients modulo a radical triangular set over the
rationals. Our modular algorithm naturally generalizes previous work from
algebraic number theory. We have implemented our algorithm using Maple's RECDEN
package. We compare our implementation with the procedure RegularGcd in the
RegularChains package.Comment: Shorter version to appear in Proceedings of SYNASC 201
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
FORM version 4.0
We present version 4.0 of the symbolic manipulation system FORM. The most
important new features are manipulation of rational polynomials and the
factorization of expressions. Many other new functions and commands are also
added; some of them are very general, while others are designed for building
specific high level packages, such as one for Groebner bases. New is also the
checkpoint facility, that allows for periodic backups during long calculations.
Lastly, FORM 4.0 has become available as open source under the GNU General
Public License version 3.Comment: 26 pages. Uses axodra
On the Extended Hensel Construction and its Application to the Computation of Real Limit Points
The Extended Hensel Construction (EHC) is a procedure which, for an input bivariate polyno- mial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. This work is motivated and illustrated by two applications. The first one is the computation of real branches of space curves. The second one is the computation of limits of real multivariate rational function. For the latter, we present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point p which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumptions on the number of variables
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