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Alfeld and Schumaker provide a formula for the dimension of the space ofpiecewise polynomial functions, called splines, of degree $d$ and smoothness$r$ on a generic triangulation of a planar simplicial complex $\Delta$, for $d\geq 3r+1$. Schenck and Stiller conjectured that this formula actually holdsfor all $d \geq 2r+1$. Up to this moment there was not known a single examplewhere one could show that the bound $d\geq 2r +1$ is sharp. However, in 2005, apossible such example was constructed to show that this bound is the bestpossible (i.e., the Alfeld-Schumaker formula does not hold if $d=2r$), exceptthat the proof that this formula actually works if $d\geq 2r+1$ has been achallenge until now when we finally show it to be true. The interesting subtleconnections with representation theory, matrix theory and commutative andhomological algebra seem to explain why this example presented such achallenge. Thus in this paper we present the first example when it is knownthat the bound $d\geq 2r+1$ is sharp for asserting the validity of theAlfeld-Schumaker formula.

From Spline Approximation to Roth's Equation and Schur Functors