Algebraic Spline Geometry: Some Remarks
In this short note we discuss some aspects of what could be called algebraic spline geometry. We concentrate on the concept of generalized Stanley–Reisner rings, namely the rings C r (Δ) of piecewise polynomial r-smooth functions on a simplicial complex Δ in ℝ d . It is well known that the geometric realization of the ordinary Stanley–Reisner ring (or face ring) C 0(Δ) reflects the structure of the simplicial complex: the irreducible components, corresponding to the maximal faces, are linear and intersect each other transversally in the pattern of the simplicial complex. We believe that the geometrical realizations of the generalized Stanley–Reisner rings behave similarly, except that the irreducible components are no longer linear and that they intersect with the appropriate multiplicity. We formulate this as a conjecture, the Local Spline Ring Conjecture, and show that it indeed holds in two very simple cases. For more complex examples, where the results are still conjectural, see N. Villamizar (Algebraic geometry for splines. Ph. D. thesis, University of Oslo, 2012, Ch. 4) for the case d = 2, r = 1.
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