Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases

Let G be the reduced Grö bner basis of a zero-dimensional ideal I ⊆ K[X, Y] of bivariate polynomials over an effective field K. Modulo suitable regularity assumptions on G and suitable precomputations as a function of G , we prove the existence of a quasi-optimal algorithm for the reduction of polynomials in K [X, Y] with respect to G . Applications include fast algorithms for multiplication in the quotient algebra A=K[X, Y] / I and for conversions due to changes of the term ordering.