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Isosingular Sets and Deflation

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Abstract

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.

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Notes

  1. Constructible algebraic sets are sets formed from algebraic sets with a finite number of Boolean operations (union, intersection, and complementation). The closure of such sets are the same in both the Zariski topology and the usual complex topology, so we draw no distinction in this article.

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Acknowledgements

The authors would like to thank Gert-Martin Greuel for sharing his knowledge regarding Thom–Boardman singularities, the Mittag-Leffler Institute (Djursholm, Sweden) for their support and hospitality, and the anonymous referees for their helpful comments.

J.D. Hauenstein was supported in part by the Mittag-Leffler Institute, and NSF grant DMS-1262428. C.W. Wampler was supported by the Mittag-Leffler Institute and NSF grant DMS-0712910.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Jonathan D. Hauenstein

  2. General Motors Research and Development, Mail Code 480-106-224, 30500 Mound Road, Warren, MI, 48090, USA

    Charles W. Wampler

Authors
  1. Jonathan D. Hauenstein
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Correspondence to Charles W. Wampler.

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Communicated by Teresa Krick.

Appendix: Foldable Stewart–Gough Platform System

Appendix: Foldable Stewart–Gough Platform System

The polynomials for the foldable Stewart–Gough platform, which are presented using the variables \([e_{1},e_{2},e_{3},e_{4},g_{1},g_{2},g_{3},g_{4}]\in {\mathbb{P}}^{7}\), are as follows. If \(r=\sqrt{3}\) and \(G=g_{1}^{2} + g_{2}^{2} + g_{3}^{2} + g_{4}^{2}\), the polynomial system is {f 1,…,f 7} defined as

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Hauenstein, J.D., Wampler, C.W. Isosingular Sets and Deflation. Found Comput Math 13, 371–403 (2013). https://doi.org/10.1007/s10208-013-9147-y

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