#
Multivariate Splines and the 4 Color Map Problem

##
Notes

The dimension statement for S13 is obtained as follows: Specify
the value and gradient of the spline at every vertex, and also
the value of the Bézier ordinate correspond to the centroid of
each triangle. This gives *3V+N* degrees of freedom. The
only remaining smoothness conditions are described by the
central quadrilaterals across each interior edge. The crucial
assumption now is that these are linearly independent. In that
case we can subtract their number from *3V+N*.
The only case known where the central smoothness conditions
are in fact linearly dependent is that of
a singular vertex.
It is an open, and apparently very difficult, question
whether there are any other triangulations where the central
smoothness conditions are in fact linearly dependent.

Click here to see a more detailed discussion of reducibility.

[07-Dec-1997]