$n$-unisolvent sets and flat incidence structures

Abstract:

For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength $n$, for short nested $n$-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are “flat” topological spaces like the Möbius strip, the cylinder, and strips of the form $I \times \mathbb {R}$, where $I$ is an interval of $\mathbb {R}$. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to $I$ in the last case. Our orthogonal arrays of strength $n$ arise from $n$-unisolvent sets of half-periodic functions, $n$-unisolvent sets of periodic functions, and $n$-unisolvent sets of functions $I\to \mathbb {R}$, respectively. Associated with every point $p$ of a nested $n$-OA, $n>1$, is a nested $(n-1)$-OA—the derived $(n-1)$-OA at the point $p$. We discover that, in our examples that arise from $n$-unisolvent sets of $n-1$ times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.