Algorithms in Local Algebra

Let k be a field, S = k[xv : v ε V] be the polynomial ring over the finite set of variables (xv : v ε V), and m = (xv : v ε V) the ideal defining the origin of Spec S. It is theoretically known (see e.g. Alonso et el., 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 Gröbner bases. Using the generalization of Moraʹs tangent cone algorithm to arbitrary term orders we give a detailed description of the necessary modifications 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 these questions were investigated in Gräbe (1991). The main algorithmic ideas described are implemented in the authorʹs REDUCE package CALI (Gräbe, 1993a).