# Algorithms for Checking Zero-Dimensional Complete Intersections

## TL;DR

This paper presents effective algorithms for verifying various types of complete intersection properties in 0-dimensional affine algebras, utilizing Fitting ideals and border bases, applicable over any base field.

## Contribution

It introduces new algorithms based on Fitting ideals and border bases to determine complete intersection properties in 0-dimensional affine algebras, extending applicability over arbitrary fields.

## Key findings

- Algorithms successfully identify complete intersection properties.
- Methods work over arbitrary base fields.
- Detection of strict complete intersections in specific ideal families.

## Abstract

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,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 characterisation 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.