Abstract: Let $\Delta$ be a triangulation of a connected region in the real plane. Let $C(r,d,\Delta)$ be the space of piecewise polynomial functions of degree $\leq d$ and smoothness $r$. A major question in Approximation Theory is to find the dimension of this space, which is not known even for the case when $d=3$ and $r=1$. Alfeld and Schumaker give a formula for this dimension, when $d\geq 3r+1$ and any $\Delta$. Using homological algebra, this problem can be translated into finding the Hilbert function of a graded module (the ``homogenization'' of $C(r,d,\Delta)$). I will discuss about this approach and about the Schenck-Stiller conjecture that says that Alfeld-Schumaker formula holds for any $d\geq 2r+1$. I will present a very recent project with Jan Minac where we prove this conjecture for a triangulation that is not trivial, in the sense that the formula does not hold if $d=2r$.