Criteria of the k-singularity and division of 1-singular systems - Karpovich, P. A. (Moscow University computational mathematics and cybernetics, 2010)
The concept of the k -singularity of systems of points in ℝ m space with l 1 metrics is studied. A system of q points is k -singular if and only if the dimensionality of the linear space of polynomials with powers no higher than the k of the columns of the matrix of pair-wise distances (element-wise multiplication) is strictly less than q . An algebraic criterion of k -singularity is obtained. The problem of dividing a system of points into subsystems that are not 1-singular is considered. An estimate of the minimum number of such subsystems is obtained.