Approximate GCD of Multivariate Polynomials

Given two polynomials F and G in R[x1, . . . , xn], we are going to find the nontrivial approximate GCD C and polynomials F , G ∈ R[x1, . . . , xn] such that ||F − CF ′|| < and ||G − CG′|| < , for some and some well defined norm. Many papers 1,2,3,5,8,10,11,13,15 have already discussed the problem in the case n = 1. Few of them 2,10,11 mentioned the case n > 1. Approximate GCD computation of univariate polynomials provides the basis for solving multivariate problem. But it is nontrivial to modify the techniques used in symbolic computation such as interpolation and Hensel lifting to compute the approximate GCD of multivariate polynomials. In section 2, we briefly review two methods 2,10 for computing GCD of polynomials with floating-point coefficients. In section 3, we focus on extending the Hensel lifting technique to polynomials with floating-point coefficients. The method involves the QR decomposition of a Sylvester matrix. We propose an efficient new algorithm which exploits the special structure of Sylvester matrix. A local optimization problem is also been proposed to improve the candidate approximate factors obtained from Hensel lifting. In order to compare the performance of the different methods, we implement all three methods in Maple. In section 4, we summarize the special problems we encounter when they are applied to polynomials with floating-point coefficients. A set of examples are given to show the efficiency and stability of the algorithms.