Computation of GCD of Sparse Multivariate Polynomials by Extended Hensel Construction

Let F(x, u1,..., uℓ) be a squarefree multivariate polynomial in main variable x and sub-variables u1,..., uℓ. We say that the leading coefficient (LC) of F is singular if it vanishes at the origin of sub-variables. A representative algorithm for nonsparse multivariate polynomial GCD is the EZ-GCD algorithm, which is based on the generalized Hensel construction (GHC). In order to apply the GHC easily, we require 1) the LC of F is nonsingular, 2) F(x, 0,..., 0) is squarefree, and 3) the initial Hensel factor of GCD is "lucky". These requirements are usually satisfied by the "nonzero substitution", i.e., to shift the origin of subvariables. However, the nonzero substitution may cause a drastic increase of the number of terms of F if F is sparse. In 1993, Sasaki and Kako proposed the extended Hensel construction (EHC) which does not perform the nonzero substitution even if the LC is singular. Using the EHC, Inaba implemented an algorithm of multivariate polynomial factorization and verified that it is very useful for sparse polynomials. In this paper, we apply the EHC for the computation of GCD of sparse multivariate polynomials. In order to find a lucky initial factor, we utilize the weighting of sub-variables, etc. Our naive implementation in Maple shows that our algorithm is comparable in performance to Maple's GCD routine based on the sparse interpolation.