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Local dimension testing via quadratic obstructions

  • Taylor Brysiewicz
  • Rainer Sinn
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Abstract

We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.

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Cite

BibTeXRIS
BibTeX
@misc{brysiewicz2026local,
  title = {{Local dimension testing via quadratic obstructions}},
  author = {Taylor Brysiewicz and Rainer Sinn},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02818},
  eprint = {2609.02818},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02818},
}
RIS
TY  - UNPB
TI  - Local dimension testing via quadratic obstructions
AU  - Taylor Brysiewicz
AU  - Rainer Sinn
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02818
UR  - https://arxiv.org/abs/2609.02818
AB  - We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.
ER  -