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Mathematics of Computation

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Dimension of mixed splines on polytopal cells

Author: Michael DiPasquale
Journal: Math. Comp. 87 (2018), 905-939
MSC (2010): Primary 13P25; Secondary 13P20, 13D02
Published electronically: September 8, 2017
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Abstract: The dimension of planar splines on polygonal subdivisions of degree at most $ d$ is known to be a degree two polynomial for $ d\gg 0$. For planar $ C^r$ splines on triangulations this formula is due to Alfeld and Schumaker; the formulas for planar splines with varying smoothness conditions across edges on convex polygonal subdvisions are due to Geramita, McDonald, and Schenck. In this paper we give a bound on how large $ d$ must be for the known polynomial formulas to give the correct dimension of the spline space. Bounds are given for central polytopal complexes in three dimensions, or polytopal cells, with varying smoothness across two-dimensional faces. In the case of tetrahedral cells with uniform smoothness $ r$ we show that the known polynomials give the correct dimension for $ d\ge 3r+2$; previously Hong and separately Ibrahim and Schumaker had shown that this bound holds for planar triangulations. All bounds are derived using techniques from computational commutative algebra.

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Additional Information

Michael DiPasquale
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Keywords: Polyhedral spline, polytopal complex, Castelnuovo-Mumford regularity, homological algebra
Received by editor(s): November 8, 2014
Received by editor(s) in revised form: January 30, 2016, July 15, 2016, and September 2, 2016
Published electronically: September 8, 2017
Additional Notes: The author was supported by National Science Foundation grant DMS 0838434 “EMSW21MCTP: Research Experience for Graduate Students”.
Article copyright: © Copyright 2017 American Mathematical Society

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