A new bound for smooth spline spaces
Commutative Algebra
2020-05-11 v3 Numerical Analysis
Algebraic Geometry
Numerical Analysis
Abstract
For a planar simplicial complex Delta contained in R^2, Schumaker proved that a lower bound on the dimension of the space C^r_k(Delta) of planar splines of smoothness r and polynomial degree at most k on Delta is given by a polynomial P_Delta(r,k), and Alfeld-Schumaker showed this polynomial gives the correct dimension when k >= 4r+1. Examples due to Morgan-Scott, Tohaneanu, and Yuan show that the equality dim C^r_k(Delta) = P_Delta(r,k) can fail when k = 2r or 2r+1. We prove that the equality dim C^r_k(Delta)= P_Delta(r,k) cannot hold in general for k <= (22r+7)/10.
Cite
@article{arxiv.1909.13399,
title = {A new bound for smooth spline spaces},
author = {Hal Schenck and Mike Stillman and Beihui Yuan},
journal= {arXiv preprint arXiv:1909.13399},
year = {2020}
}
Comments
v1: 4 pages, 1 figure. v2: minor stylistic changes. v3: added additional algebraic background