Fast reduction of bivariate polynomials with respect to sufficiently regular Gröbner bases

Joris van der Hoeven (1) , Robin Larrieu (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Let G be the reduced Gröbner 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.

Domaines: Calcul formel [cs.SC]; Complexité [cs.CC]; Logiciel mathématique [cs.MS]; Anneaux et algèbres [math.RA]

Joris van der Hoeven, Robin Larrieu. Fast reduction of bivariate polynomials with respect to sufficiently regular Gröbner bases. 2018. ⟨hal-01702547v2⟩