Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x

Abstract

In this paper we present two new methods for computing thesubresultant polynomial remainder sequence (prs) of two polynomials f, g ∈ Z[x]. We are now able to also correctly compute the Euclidean and modifiedEuclidean prs of f, g by using either of the functions employed by ourmethods to compute the remainder polynomials.Another innovation is that we are able to obtain subresultant prs’s inZ[x] by employing the function rem(f, g, x) to compute the remainderpolynomials in [x]. This is achieved by our method subresultants_amv_q(f, g, x), which is somewhat slow due to the inherent higher cost of com-putations in the field of rationals.To improve in speed, our second method, subresultants_amv(f, g,x), computes the remainder polynomials in the ring Z[x] by employing thefunction rem_z(f, g, x); the time complexity and performance of thismethod are very competitive.Our methods are two different implementations of Theorem 1 (Section 3),which establishes a one-to-one correspondence between the Euclidean andmodified Euclidean prs of f, g, on one hand, and the subresultant prs of f, g,on the other.By contrast, if – as is currently the practice – the remainder polynomi-als are obtained by the pseudo-remainders function prem(f, g, x) 3 , thenonly subresultant prs’s are correctly computed. Euclidean and modified Eu-clidean prs’s generated by this function may cause confusion with the signsand conflict with Theorem 1.ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.