Strang’s conjecture for bivariate C1 cubic splines

For every triangulation $\Delta$ of a planar polygonal domain, let $S_3^1(\Delta)$ denote the space of functions that are polynomial of total degree at most $3$ on each triangle of $\Delta$ and are globally $C^1$ . If $f_0(\Delta)$ is the number of vertices and $f_0^\circ(\Delta)$ is the number of interior vertices, then $\dim S_3^1(\Delta)=3f_0(\Delta)-f_0^\circ(\Delta)+1$ .

References

Primary source

arXiv

Additional references

A symmetric counterexample to Strang's conjecture for bivariate $C^1$ cubic splines on triangulations — arXiv — Pratyush Potu

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

An unrefereed preprint reports a symmetric counterexample, so the conjectured dimension formula is claimed false, but the result has not been independently verified.

Strang’s conjecture predicts a dimension formula for bivariate $C^1$ cubic spline spaces on triangulations. The reported examples claim that the formula fails even for highly symmetric, nondegenerate triangulations.

October 6, 2026 symmetric counterexample

Pratyush Potu reports an equilateral-triangle triangulation invariant under $D_3$ , with no collinear edges sharing a vertex, for which $\dim S_3^1=41$ while the conjectured lower-bound expression is $40$ . An earlier preprint reports a nonsymmetric exceptional realization with dimension $34$ instead of $33$ . A separate preprint claims the conjecture under strong collapsibility, but only under that additional hypothesis.

Current status (as of October 2026): The conjecture is claimed false by two unrefereed counterexamples, while the restricted positive result does not settle the general case.

Sources

No solutions have been posted yet.