Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

Abstract 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 \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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