Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

@article{Chistov2003VandermondeMN,
  title={Vandermonde Matrices, NP-Completeness, and Transversal Subspaces},
  author={Alexander L. Chistov and Herv{\'e} Fournier and Leonid Gurvits and Pascal Koiran},
  journal={Foundations of Computational Mathematics},
  year={2003},
  volume={3},
  pages={421-427}
}
Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of Kn with the following transversality property: any linear subspace of Kn 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 m and a n × m matrix A with entries in Z, decide whether there exists a n × n sub-determinant of A which is equal to zero. 
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