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Let be the reduced Gröbner basis of a zero-dimensional ideal of bivariate polynomials over an effective field . Modulo suitable regularity assumptions on and suitable precomputations as a function of , we prove the existence of a quasi-optimal algorithm for the reduction of polynomials in with respect to . Applications include fast algorithms for multiplication in the quotient algebra and for conversions due to changes of the term ordering.

**Authors:**

**Keywords: **polynomial reduction, algorithm, complexity,
Gröbner basis