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The space S13 is particularly interesting because it has the smallest possible polynomial degree for which the dimension is known to be larger than the number of vertices in the triangulation.

The question of the dimension of *S13* is probably
the most famous unsolved problem in this flavor of
multivariate spline theory.

Of course, for any **particular** triangulation one can
analyze an appropriate linear system and compute the
dimension for that case. But we would like to have a formula
that gives the dimension without having to solve a (large)
linear system.

In general, everything gets easier as the polynomial degree
increases. Dimension formulas for S1d are known if *d
> 3*, and for Srd (where the spline is r times
differentiable) exact dimension formulas are known when *
d > d3 +1*, and the generic dimension is known when
*d = 3r+1.*

When *d> 3r+1* the smoothness conditions
effectively decouple and the analysis can be done locally.

A special case of the spline spaces *Srd* are
triangle based *finite elements*.

[28-Apr-1997]