Computing generic bivariate Gröbner bases with Mathemagix

Abstract

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, 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.