A smoothness test for higher codimensions

TL;DR

This paper introduces an efficient, parallel algorithmic smoothness test for algebraic varieties in higher codimensions, avoiding large Jacobian minors, and combines it with the Jacobian criterion for practical use.

Contribution

It presents a novel smoothness test based on Hironaka's ideas, implemented in Singular, that is more practical for high codimension varieties than traditional Jacobian methods.

Findings

The new test is faster and uses less memory than the Jacobian criterion.

The hybrid method effectively combines strengths of both approaches.

Implementation demonstrates practical advantages in algebraic geometry applications.

Abstract

Based on an idea in Hironaka's proof of resolution of singularities, we present an algorithmic smoothness test for algebraic varieties. The test is inherently parallel and does not involve the calculation of codimension-sized minors of the Jacobian matrix of the variety. We also describe a hybrid method which combines the new method with the Jacobian criterion, thus making use of the strengths of both approaches. We have implemented all algorithms in the computer algebra system Singular, and compare the different approaches with respect to timings and memory usage. The test examples originate from questions in algebraic geometry, where the use of the Jacobian criterion is impractical due to the number and size of the minors involved.