Resultant Based Methods Computing the Greatest Common Divisor of Several Polynomials

Abstract

In this paper we develop two resultant based methods for the computation of the greatest common divisor (GCD) of many polynomials. Let S be the resultant Sylvester's matrix of the polynomials. The application of classical LU and QR factorization to S for the computation of its GCD has an inappropriate complexity of order O(n4). We modified matrix S to S* such that the rows with non-zero elements under the main diagonal, at every column, are gathered together. We constructed modified versions of the LU and QR procedures which lead to the computation of the GCD of S* in O(n3) floating point operations. Both methods are tested for several sets of polynomials and tables summarizing all the achieved results are given