Gröbner bases as characteristic sets (Q2717165)

Let \(\sigma\) be a term ordering on the polynomial ring \(A := k[x_1, \dots, x_n]\), then it is possible to define a pre-order on \(A\) (called ``ranking pre-order''), in which, given \( f, g \in A\), \(f\) has lower rank than \(g\) if the maximal monomial of \(f\) is less than the maximal monomial of \(g\). NEWLINENEWLINENEWLINEGiven \(\sigma\), an ``autoreduced subset'' of \(A\) is a finite subset of \(A\) in which any element \(g\) of \(G\) is reduced with respect to \(G \setminus \{g \}\). The above ranking pre-order can be extended to autoreduced subsets of \(A\). In particular the author shows that for any given ideal \(I\) of \(A\) there exists an autoreduced subset of \(I\) of lowest rank, called a ``characteristic set'' of \(I\). NEWLINENEWLINENEWLINEThe author shows that a characteristic set of \(I\) is precisely a reduced Gröbner basis of \(I\); hence it follows that the Wu-Ritt-Kolchin algorithm allows to compute Gröbner bases. NEWLINENEWLINENEWLINEFinally, the ranking pre-order induces a linear pre-order on the set of all monomial ideals and it induces a linear pre-order on the Hilbert scheme of all homogeneous ideals in \(A\) with the same Hilbert polynomial.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].