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 : = ⟨ A , B ⟩ 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.