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Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

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Abstract

Let \(A, B \in \mathbb {K} [X, Y]\) be two bivariate polynomials over an effective field \(\mathbb {K}\), and let G be the reduced Gröbner basis of the ideal \(I :=\langle A, B \rangle \) 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 \(P \in \mathbb {K} [X, Y]\) 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 \(\mathbb {A} :=\mathbb {K} [X, Y] / \langle A, B \rangle \), 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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Notes

  1. The results from [18] actually apply for more general types of supports, but this will not be needed in this paper.

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Acknowledgements

We thank Vincent Neiger for a remark that simplified Algorithm 6. We also thank the anonymous referees for helpful comments and suggestions.

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  1. Laboratoire d’informatique de l’École polytechnique, LIX, UMR 7161, CNRS & École polytechnique, Campus de l’École polytechnique, 1 rue Honoré d’Estienne d’Orves, Bâtiment Alan Turing, CS35003, 91120, Palaiseau, France

    Joris van der Hoeven & Robin Larrieu

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  1. Joris van der Hoeven
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  2. Robin Larrieu
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Correspondence to Joris van der Hoeven.

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van der Hoeven, J., Larrieu, R. Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals. AAECC 30, 509–539 (2019). https://doi.org/10.1007/s00200-019-00389-9

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