Schumaker’s lower-bound equality conjecture for C^1 cubic spline spaces

The conjecture asks whether Schumaker’s lower bound always equals the dimension of planar $C^1$ cubic spline spaces, including the correction term $\sigma$. It arose from work of Strang and Schumaker and remained open for arbitrary triangulations.

Known results

• Schumaker (1979, 1984) established the lower bound and the local correction for singular interior four-stars.
• Billera (1988) proved equality for generic triangulations in the case $(r,k)=(1,3)$.
• Schenck, Stillman, and Yuan (2019) established broad high-degree results and documented failures in other low-degree regimes.

2026 counterexample claim

The preprint Geometry-dependent rank defect in $C^1$ cubic spline space claims that an $18$-triangle family has $\dim S_3^1=34$ at $t=1/5$, while $P_{\mathcal T}(1,3)=33$ and $\sigma=0$. If correct, this is a nonsingular global rank defect and disproves universal equality; the calculation remains unverified.

Current status (as of September 2026): The conjecture has not been verified as false; the claimed $18$-triangle counterexample remains unconfirmed, so the exact arbitrary-triangulation case is unresolved.