Resolve Strang’s dimension conjecture

Prove or disprove Strang’s conjecture that the dimension of the space of continuously differentiable bivariate splines of degree at most k on a triangulation of a simply connected polygonal domain equals the combinatorial Strang number, with the appropriate correction by the number of singular vertices.

Background

Strang’s conjecture concerns the dimension of the space of bivariate C1 splines of degree at most k on a triangulation of a connected polygonal region with no holes. The proposed dimension is expressed in terms of the numbers of triangles, interior edges, and interior vertices, with a correction term for singular vertices.

The paper notes that the conjecture has been proved for k equal to 4 and for k at least 5, while a counterexample exists for k equal to 2. The paper proves a positive result for k=2 under strong collapsibility and a non-collinearity condition, so the conjecture remains unresolved in general.

References

It was conjectured by Strang that the dimension of $S_hk(\Th)$ is given by eq:strangdim.

eq:strangdim:

Shk=(k+22)NT−(2k+1)Ne˚+3Nv˚\mathcal{S}_h^k= \binom{k+2}{2}N_T-(2k+1)\mathring{N_e} +3\mathring{N_v}

— Strang's Conjecture: Positive Result on Strong Collapsible Complexes and a Code to Check the Conjecture  (2608.24120 - Romero et al., 25 Aug 2026) in Section 1, Introduction

The case $k=3$ remains open.

— Strang's Conjecture: Positive Result on Strong Collapsible Complexes and a Code to Check the Conjecture  (2608.24120 - Romero et al., 25 Aug 2026) in Section 1, Introduction