Criteria of the k-singularity and division of 1-singular systems

Abstract

The concept of the k-singularity of systems of points in ℝm space with l1 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.