Computing a Maximal Independent Set Modulo an Ideal and a Gröbner Basis of the Ideal Simultaneously

To solve problems on a positive-dimensional ideal I ⊂ k[X], a maximal independent set U ⊂ X modulo I and a Gröbner basis of Ie, where Ie is the extension of I to k(U)[V] (V := X\U), are widely used. As far as we know, they are usually computed separately, i.e., U is calculated first and the Gröbner basis is computed after U is obtained. In this paper, we present an efficient algorithm for computing a maximal independent set U modulo I and a Gröbner basis of Ie simultaneously. Different from computing them separately, the algorithm takes full advantage of the polynomial information throughout the Gröbner basis computation to obtain U as soon as possible, hence it significantly improves the computing efficiency.