Algorithms in Local Algebra

Let k be a eld, S = k[xv : v 2 V ] be the polynomial ring over the nite set of variables (xv : v 2 V), and m = (x v : v 2 V) the ideal dening the origin of Spec S. It is theoretically known (see e.g. Alonso et al. 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computations in the local ring Sm but also for term orders of mixed type with standard bases replacing Gr obner bases. Using the generalization of Mora's tangent cone algorithm to arbitrary term orders we give a detailed description of the necessary modications and restrictions. In a second part we discuss a generalization of the deformation argument for standard bases and independent sets to term orders of mixed type. For local term orders they were investigated in (Gr abe 1991). The main algorithmic ideas described below are implemented in the author's REDUCE package CALI (Gr abe 1993a).