Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

  • Joris van der HoevenEmail author
  • Robin Larrieu
Original Paper


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 ABP. 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.


Polynomial reduction Gröbner basis Complexity Algorithm 

Mathematics Subject Classification




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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire d’informatique de l’École polytechnique, LIX, UMR 7161CNRS & École polytechniquePalaiseauFrance

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