Regularity of Mixed Spline Spaces

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We derive bounds on the regularity of the algebra Cα(P) of mixed splines over a central polytopal complex P³. As a consequence we bound the largest integer d (the postulation number) for which the Hilbert polynomial HP(Cα(P),d) disagrees with the Hilbert function HF(Cα(P),d)= Cα(P)d. The polynomial HP(Cα(P),d) has been computed in [DiPasquale 2014], building on [McDonald-Schenck 09] and [Geramita-Schenck 98]. Hence the regularity bounds obtained indicate when a known polynomial gives the correct dimension of the spline space Cα(P)d. In the simplicial case with all smoothness parameters equal, we recover a bound originally due to [Hong 91] and [Ibrahim and Schumaker 91].