Abstract: Let $\Delta$ be a triangulation of a topological open disk in the real plane. Let $r$ and $d$ be two positive integers. On this region one defines a piecewise $C^r$ function, such that on each triangle the function is given by a polynomial in two variables of degree $\leq d$. The set of these functions forms a finite dimensional vector space, and one of the major questions in Approximation Theory is to find the dimension of this space. It was conjectured that for $d\geq 2r+1$, this dimension is given by a precise formula that depends on the combinatorial information of the simplicial complex $\Delta$, and on the local geometric data. The conjecture is very difficult, and trying to prove it for the simplest nontrivial example has been a challenge for about 10 years. Jan Minac and myself answered this question by the means of Commutative Algebra, showing also that a direct approach to solve this conjecture for this particular example leads to difficult questions in Matrix Theory, such as the LU-decomposition of an invertible matrix. In this talk I am presenting an overview of these problems. The talk is accessible to graduate students.