Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

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 := hA, Bi 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 ∊ K[X, Y ] modulo G, where “quasi-optimal” is meant in terms of the size of the input A, B, P. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra A:= K[X, Y ]/〈A, B〉, both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.