2,204 research outputs found
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
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
The Newton Polytope of the Implicit Equation
We apply tropical geometry to study the image of a map defined by Laurent
polynomials with generic coefficients. If this image is a hypersurface then our
approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure
On Modular Inverses of Cyclotomic Polynomials and the Magnitude of their Coefficients
Let p and r be two primes and n, m be two distinct divisors of pr. Consider
the n-th and m-th cyclotomic polynomials. In this paper, we present lower and
upper bounds for the coefficients of the inverse of one of them modulo the
other one. We mention an application to torus-based cryptography.Comment: 21 page
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