Computing generic bivariate Gröbner bases with Mathemagix

Robin Larrieu (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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 lexico-graphic order. Assuming A and B sufficiently generic, G admits a so-called concise representation that helps computing normal forms more efficiently [7]. Actually, given this concise representation, a polynomial P ∈ K[X, Y ] can be reduced modulo G with quasi-optimal complexity (in terms of the size of the input A, B, P). Moreover, the concise representation can be computed from the input A, B with quasi-optimal complexity as well. The present paper reports on an efficient implementation for these two tasks in the free software Mathemagix [10]. This implementation is included in Mathemagix as a library called larrix.

Domaines: Calcul formel [cs.SC]

Robin Larrieu. Computing generic bivariate Gröbner bases with Mathemagix. ACM Communications in Computer Algebra, 2019, 53 (2), pp.41-44. ⟨10.1145/3371991.3371994⟩. ⟨hal-02387947⟩