Probability and interpolation

Abstract:

An $m \times n$ matrix $E$ with $n$ ones and $(m - 1)n$ zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, $E$ is singular with probability that converges to one if $m$, $n \to \infty$. Previously, this was known only if $m \geqslant (1 + \delta )n/\log n$. For constant $m$ and $n \to \infty$, the probability is asymptotically at least $\tfrac {1} {2}$.