La queste del saint Gr a(AL): A computational approach to local algebra

Abstract

We show how, by means of the Tangent Cone Algorithm, the basic functions related to the maximal ideal topology of a local ring can be effectively computed in the situations of geometrical significance, i.e.: (1) localizations of coordinate rings of a variety at the prime ideal defining a subvariety, (2) rings of algebraic formal power series rings. In particular we show how the method of “associated graded rings” can be turned into an effective tool to compute local algebraic invariants of varieties.