Computing generic bivariate Gröbner bases with Mathemagix
Robin Larrieu · ACM communications in computer algebra · 2019
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.