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Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

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Abstract

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of K n with the following transversality property: any linear subspace of K n of dimension n–r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n \leq m and a n × m matrix A with entries in Z, decide whether there exists an n × n subdeterminant of A which is equal to zero.

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Authors and Affiliations

  1. Saint Petersburg Institute for Informatics and Automation of the Academy of Sciences of Russia, Saint Petersburg 199178, Russia

    Alexander Chistov

  2. Université de Versailles Saint-Quentin en Yvelines, 45, avenue des États-Unis, 78035 Versailles Cedex, France

    Hervé Fournier

  3. Los Alamos National Laboratory, CCS-3, Los Alamos, NM 87545, USA

    Leonid Gurvits

  4. LIP, Ecole Normale Supérieure de Lyon, 46 allee d’Italie, 69364 Lyon Cedex 07, France

    Pascal Koiran

Authors
  1. Alexander Chistov
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  2. Hervé Fournier
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  3. Leonid Gurvits
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  4. Pascal Koiran
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Corresponding authors

Correspondence to Alexander Chistov, Hervé Fournier, Leonid Gurvits or Pascal Koiran.

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Chistov, A., Fournier, H., Gurvits, L. et al. Vandermonde Matrices, NP-Completeness, and Transversal Subspaces. Found Comput Math 3, 421–427 (2003). https://doi.org/10.1007/s10208-002-0076-4

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  • DOI: https://doi.org/10.1007/s10208-002-0076-4

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