Algebraic Factorization and GCD Computation
نویسنده
چکیده
This chapter describes several algorithms for factorization and GCD computation of polynomials over algebraic extension fields. These algorithms are common in using the characteristic set method introduced in the previous chapters. Some performance comparisons between these algorithms are reported. Applications include geometry theorem proving, irreducible decomposition of algebraic variaities, implicitization of parametric equations and verification of geometric conditions.
منابع مشابه
Computing Greatest Common Divisors and Factorizations
In a quadratic number field Q(V~D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains Euclidean algorithms are known. It was shown by Cohn [5] that for D < —19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does no...
متن کاملComputing Greatest Common Divisors and Factorizations in Quadratic Number Fields*
In a quadratic number field Q(√ D ), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. It was shown by Cohn [5] that for D ≤ – 19 even remainder sequences with possibly non-decreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does...
متن کاملTwo Families of Algorithms for Symbolic Polynomials
We consider multivariate polynomials with exponents that are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and projection methods. The first family of algorithms uses the algebraic independence of x, x, x 2 , x, etc, to solve related problems with more indet...
متن کاملChapter 1 Regularization and Matrix Computation in Numerical Polynomial Algebra
Numerical polynomial algebra emerges as a growing field of study in recent years with a broad spectrum of applications and many robust algorithms. Among the challenges in solving polynomial algebra problems with floating-point arithmetic, difficulties frequently arise in regularizing ill-posedness and handling large matrices. We elaborate regularization principles for reformulating the illposed...
متن کاملThe University of Western Ontario the School of Graduate And
The factor refinement principle turns a partial factorization of integers (or polynomials) into a more complete factorization represented by basis elements and exponents, with basis elements that are pairwise coprime. There are lots of applications of this refinement technique such as simplifying systems of polynomial inequations and, more generally, speeding up certain algebraic algorithms by ...
متن کامل