  Semialgebraic splines are functions that are piecewise polynomial with
respect to a cell decomposition into sets defined by polynomial inequalities.
We study bivariate semialgebraic splines, formulating spaces of semialgebraic
splines in terms of graded modules. We compute the dimension of the space of
splines with large degree in two extreme cases when the cell decomposition has
a single interior 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. We also study examples of the Hilbert function and polynomial in
cases of a single vertex where the curves do not satisfy either of these
extremes.
