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On the genericity of the modular polynomial GCD algorithm

In this paper we study the generic setting of the modular GCD algorithm. We develop the algorithm for multivariate polynomials over Euclidean domains which have a special kind of remainder function. Details for the parameterization and generic Maple code are given. Applying this generic algorithm to a GCD problem in Z/(p)[t][x] where p is small yields an improved asymptotic performance over the usual approach, and a very practical algorithm for polynomials over small finite fields.