Semialgebraic splines

Semialgebraic splines are functions that are piecewise polynomial 
with respect to a cell decomposition of their domain into sets 
defined by polynomial inequalities. We study bivariate semialgebraic 
splines, computing the the dimension of the space of splines with 
large degree in two extreme cases when the cell decomposition has 
a single internal vertex. First, when the forms defining the edges 
span a two-dimensional space of forms of degree n---then the curves 
they define meet in n^2 points in the complex projective plane. 
In the other extreme, the curves have distinct slopes at the vertex 
and do not simultaneously vanish at any other point.

This is joint work with Michael DiPasquale and Lanyin Sun