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A Lower Bound for Splines on Tetrahedral Vertex Stars / Michael DiPasquale, Nelly Villamizar

SIAM Journal on Applied Algebra and Geometry, Volume: 5, Issue: 2, Pages: 250 - 277

Swansea University Author: Nelly Villamizar

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DOI (Published version): 10.1137/20m1341118

Abstract

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the d...

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Published in: SIAM Journal on Applied Algebra and Geometry
ISSN: 2470-6566
Published: Society for Industrial & Applied Mathematics (SIAM) 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa56990
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Abstract: A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra|we call theseclosed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of splines whose derivatives up to order r are continuous on closed vertex stars of degree at least 3r + 2. We show that this formula is a lower bound on the dimension of splines of degree at least (3r + 2)/2. Our proof uses apolarity and thevso-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most (3r + 1)/2 on a generic closed vertex star are global polynomials.
Keywords: spline functions, apolarity, fat point ideals, Waldschmidt constant
College: College of Science
Issue: 2
Start Page: 250
End Page: 277