Elsevier

Journal of Symbolic Computation

Volume 86, May–June 2018, Pages 153-165
Journal of Symbolic Computation

A smoothness test for higher codimensions

https://doi.org/10.1016/j.jsc.2017.05.001Get rights and content
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Abstract

Based on an idea in Hironaka's proof of resolution of singularities, we present an algorithm for determining smoothness of algebraic varieties. The algorithm 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. We 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.

Keywords

Singularities
Algorithmic smoothness test
Hironaka resolution of singularities
Unprojection
Parallel computation in algebraic geometry

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This research project was partially supported within DFG SPP1489 and by NTH Bottom Up project ‘Experimental Methods in Computer Algebra’.