Let Delta be a triangulation of a simply connected polygonal domain Omega⊂R^2, let f_1^circ and f_0^circ denote the numbers of interior edges and interior vertices of Delta, and define S_k^1(Delta)=s∈ C^1(Omega):s|_T∈P_k(T) for every triangle T∈Delta. Strang's conjecture asserts the dimension formula dim S_k^1(Delta)=binomk+22+f_1^circ binomk2-f_0^circ binomk-22, with binomn2=0 when n<2, for the relevant degrees k.

Strang’s conjecture

Let $\Delta$ be a triangulation of a simply connected polygonal domain $\Omega\subset\mathbb{R}^2$ , let $f_1^\circ$ and $f_0^\circ$ denote the numbers of interior edges and interior vertices of $\Delta$ , and define $S_k^1(\Delta)=\{s\in C^1(\Omega):s|_T\in\mathbb{P}_k(T)\text{ for every triangle }T\in\Delta\}$ . Strang's conjecture asserts the dimension formula $\dim S_k^1(\Delta)=\binom{k+2}{2}+f_1^\circ\binom{k}{2}-f_0^\circ\binom{k-2}{2}$ , with $\binom{n}{2}=0$ when $n<2$ , for the relevant degrees $k$ .

References

Primary source

arXiv

Additional references

Strang's Conjecture: Positive Result on Strong Collapsible Complexes and a Code to Check the Conjecture — arXiv

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

The general dimension formula is not proved: many cases are settled, but the cubic case and some low-degree cases remain open, with a new test covering only a restricted class.

Strang proposed the dimension formula in 1974, initially for the case $r=1$ , $k=3$ . The broader conjecture asks when the expected dimension formula holds for $C^1$ polynomial splines on triangulations; it is not universally valid in low degrees.

Known results

Schumaker proved the conjectured expression is always a lower bound.
Alfeld–Schumaker proved equality for $k\geq 4r+1$ ; Hong improved this to $k\geq 3r+2$ .
Billera proved the $r=1$ , $k=3$ case for generic triangulations.
Morgan–Scott gave a counterexample at $r=1$ , $k=2$ ; equality is known for all triangulations when $k=4$ or $k\geq 5$ , while $k=3$ remains open.

August 2026 positive result

An unrefereed preprint gives an equivalent surjectivity criterion, supplies code for computational checks, and proves the $k=2$ case for strongly collapsible complexes satisfying an additional non-collinearity condition. This is a restricted positive result, not a resolution of the general conjecture.

Current status (as of September 2026): The formula is settled for $k=4$ and $k\geq 5$ , fails in general for $k=2$ , and remains open for $k=3$ ; the August preprint adds only an unverified restricted case and a computational criterion.

Sources

No solutions have been posted yet.