Three topics in multivariate spline theory

We examine three topics at the interface between spline theory and algebraic geometry. In the first part, we show how the concept of domain points can be used to give an original explanation of Dehn–Sommerville equations relating the numbers of i-faces of a simplicial polytope in R, i = 0. . . . , n − 1. In the second part, we echo some joint works with T. Sorokina and with P. Clarke on computational methods that generate formulas for the dimensions of spline spaces S d(∆n) of degree ≤ d and smoothness r over a fixed simplicial partition ∆n in R. It exploits the specific form of the generating function ∑ d≥0 dimS d(∆n)z — the so-called Hilbert series. In the third and final part, we state that Schumaker’s conjecture about bivariate interpolation at subsets of domain points holds up to degree d = 17, with extensions to the trivariate and quadrivariate cases. We also reformulate the conjecture in three different ways, especially as a question about a certain bivariate Vandermonde matrix being a P -matrix.