From spline approximation to Roth’s equation and Schur functors

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Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ, for d ≥ 3r + 1. Schenck and Stiller conjectured that this formula actually holds for all d ≥ 2r + 1. Up to this moment there was not known a single example where one could show that the bound d ≥ 2r + 1 is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld–Schumaker formula does not hold if d = 2r), except that the proof that this formula actually works if d ≥ 2r + 1 has been a challenge until now when we finally show it to be true. The interesting subtle connections with representation theory, matrix theory and commutative and homological algebra seem to explain why this example presented such a challenge. Thus in this paper we present the first example when it is known that the bound d ≥ 2r + 1 is sharp for asserting the validity of the Alfeld–Schumaker formula.