Bivariate Semialgebraic Splines

We consider bivariate splines over partitions defined by 
arcs of irreducible algebraic curves.  We compute the 
dimension of the space of semialgebraic splines in two 
extreme cases.  If the forms defining the edges span a 
three-dimensional space of forms of degree $n$, then we 
show that the dimensions can be reduced to the linear case.  
If the partition is sufficiently generic, we give a 
formula for the dimension of the spline space in large 
degree and bound how large the degree must be for the 
formula to be correct.  We also study the dimension of 
the spline space in some examples where the curves do 
not satisfy either extreme.  The results are derived 
using commutative and homological algebra.  This is joint
work with Michael DiPasquale