Computing generic bivariate Grobner bases with MATHEMAGIX

Let A, B is an element of K[X, Y] be two bivariate polynomials over an effective field K, and let G be the reduced Grobner basis of the ideal I := hA, Bi 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 is an element of 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.