From Spline Approximation to Roth's Equation and Schur Functors

TL;DR

This paper proves that the Alfeld-Schumaker formula for spline space dimensions holds precisely when the degree exceeds twice the smoothness, resolving a longstanding conjecture and demonstrating the sharpness of the bound with a concrete example.

Contribution

The paper confirms the conjecture that the formula holds for all degrees at least 2r+1 and provides the first explicit example showing the bound is sharp.

Findings

The Alfeld-Schumaker formula is valid for all d ≥ 2r+1.

The bound d ≥ 2r+1 is proven to be sharp.

An explicit example demonstrates the sharpness of the bound.

Abstract

Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\Delta$, for $d \geq 3r+1$. Schenck and Stiller conjectured that this formula actually holds for all $d \geq 2r+1$. Up to this moment there was not known a single example where one could show that the bound $d\geq 2r +1$ is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld-Schumaker formula does not hold if $d=2r$), except that the proof that this formula actually works if $d\geq 2r+1$ has been a challenge until now when we finally show it to be true. The interesting subtle connections with representation theory, matrix theory and commutative and homological algebra seem to explain why this example presented such a challenge. Thus in this paper we present the first example when it is known that the bound $d\geq 2r+1$ is sharp for asserting the validity of the Alfeld-Schumaker formula.