Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

Abstract

Let $$A, B \in \mathbb {K} [X, Y]$$ be two bivariate polynomials over an effective field $$\mathbb {K}$$ , and let G be the reduced Grobner basis of the ideal $$I :=\langle A, B \rangle $$ 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 $$P \in \mathbb {K} [X, Y]$$ 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 $$\mathbb {A} :=\mathbb {K} [X, Y] / \langle A, B \rangle $$ , both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.