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Article Dans Une Revue ACM Communications in Computer Algebra Année : 2019

Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

Résumé

Let A, B ∈ K[X, Y ] be two bivariate polynomials over an effective field K, and let G be the reduced Gröbner basis of the ideal I := A, B generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, we design a quasi-optimal algorithm for the reduction of P ∈ K[X, Y ] modulo G, where "quasi-optimal" is meant in terms of the size of the input A, B, P. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra A := K[X, Y ]/ A, B , both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.
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Dates et versions

hal-02350401 , version 1 (23-11-2020)

Identifiants

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Joris van der Hoeven, Robin Larrieu. Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals. ACM Communications in Computer Algebra, 2019, 52 (3), pp.55-58. ⟨10.1145/3313880.3313882⟩. ⟨hal-02350401⟩
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