Algorithms for checking zero-dimensional complete intersections

Abstract

Given a 0-dimensional affine $K$-algebra $R=K[x_1,\dots ,x_n]∕I$, where $I$ is an ideal in a polynomial ring $K[x_1,\dots ,x_n]$ over a field $K$, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether $R$ is a complete intersection at a maximal ideal, whether $R$ is locally a complete intersection, and whether $R$ is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of $I$ form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field $K$. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.