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We present an algorithm to compute the primary decomposition of any ideal in a polynomial ring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic… (More)

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems , and give efficient algorithms for computing precisely how… (More)

Algorithms are developed that adopt a novel implicit representation for multi-variate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and… (More)

Given a squarefree polynomial P ∈ k 0 [x, y], k 0 a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate information concerning the Galois group of P over k 0… (More)

Atntract \Ve discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factoriza-tion, to find the roots of a system of multivariate polynomial equations. The principal contribution of this paper is to show how to reduce thr multivariate problem to a univariate problem, even in the raw of multiple roots, in a… (More)

<bold>This paper reports ongoing research at the IBM Research Center on the development of a language with extensible parameterized types and generic operators for computational algebra. The language provides an abstract data type mechanism for defining algorithms which work in as general a setting as possible. The language is based on the notions of… (More)

This paper considers the problem of factoring polynomials over a variety of domains. We first describe the current methods of factoring polynomials over the integers, and extend them to the integers mod p. We then consider the problem of factoring over algebraic domains. Having produced several negative results, showing that, if the domain is not properly… (More)

This paper presents a new, simple, and efficient algorithm for factoring polynomials in several variables over an algebraic number field. The algorithm is then used iteratively, to construct the splitting field of a polynomial over the integers. Finally the factorization and splitting field algorithms are applied to the problem of determining the… (More)

It is proved that, under the usual restrictions, the denominator of the integral of a purely logarithmic function is the expected one, that is, all factors of the denominator of the integrand have their multiplicity decreased by one. Furthermore, it is determined which new logarithms may appear in the integration.

While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators it is necessary to introduce and, in short, how one can… (More)