Computing greatest common divisors and squarefree decompositions through matrix methods: The parametric and approximate cases

Abstract

Barnett’s method through Bezoutians is a purely linear algebra method allowing to compute the degree of the greatest common divisor of several univariate polynomials in a very compact way. Two different uses of this method in computer algebra are introduced here. Firstly, we describe an algorithm for parameterizing the greatest common divisor of several polynomials in K [ x , y ] , being x a parameter taking values in an real field K . Secondly, we consider the problem of computing the approximate greatest common divisor with limited accuracy for several univariate polynomials following Corless et al. [R.M. Corless, P.M. Gianni, B.M. Trager, S. Watt, The singular value decomposition for polynomial systems, in: ACM International Symposium on Symbolic and Algebraic Computation, 1995, pp. 195–207]. Given a family of polynomials whose coefficients are imperfectly known, we describe an algorithm for computing their approximate greatest common divisor by using, as main tools, Barnett’s method and singular value decomposition computations. Furthermore, we show how to use this algorithm in order to obtain the approximate squarefree decomposition of a given polynomial with imperfectly known coefficients.