Complexity of Zero-dimensional Gröbner Bases
Résumé
In this paper, it is shown that the Gröbner basis (for any monomial ordering) of a zero-dimensional ideal may be computed within a bit complexity which is essentially polynomial in $D^n$ where $n$ is the number of unknowns and $D$ the mean value of the degrees of input polynomials. Therefore, if all input polynomials have the same degree, this complexity is polynomial in the maximum of the input size and of the output size, for almost all inputs. The proof is designed in order that it may be generalized to the case of an ideal generated by a regular sequence and for the degree reverse lexicographic ordering, when the last variables are in generic position for the ideal. But one step of the proof is yet lacking in this case.