# Testing zero-dimensionality of varieties at a point

## TL;DR

This paper introduces effective algorithms using comprehensive Gröbner systems to test zero-dimensionality of algebraic varieties at a point, analyze deformations of singularities, and compute local dimensions, with applications to parameter space stratification.

## Contribution

It presents novel algorithms leveraging comprehensive Gröbner systems for zero-dimensionality testing and local dimension computation at a point, including parameter space decomposition.

## Key findings

- Algorithms successfully determine zero-dimensionality at a point.
- Methods enable analysis of deformations of hypersurface singularities.
- Parameter space can be decomposed into strata based on local dimension.

## Abstract

Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing local dimensions are also described. For the case where a given ideal contains parameters, the proposed algorithms can output in particular a decomposition of a parameter space into strata according to the local dimension at a point of the associated varieties. The key of the proposed algorithms is the use of the notion of comprehensive Gr\"obner systems.