Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2018

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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Commentaire Proof-of-concept implementation
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Dates et versions

hal-01770408 , version 1 (19-04-2018)
hal-01770408 , version 2 (01-02-2019)
hal-01770408 , version 3 (28-11-2020)

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  • HAL Id : hal-01770408 , version 1

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Joris van der Hoeven, Robin Larrieu. Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals. 2018. ⟨hal-01770408v1⟩
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