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Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach

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Abstract

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight of different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Fröberg–Iarrobino conjecture, and the weak Lefschetz property.

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Authors and Affiliations

  1. Inria Sophia Antipolis Méditerranée, 2004 route des Lucioles, B.P. 93, 06902, Sophia Antipolis, France

    Bernard Mourrain

  2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040, Linz, Austria

    Nelly Villamizar

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  1. Bernard Mourrain
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  2. Nelly Villamizar
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Correspondence to Nelly Villamizar.

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Mourrain, B., Villamizar, N. Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach. Math.Comput.Sci. 8, 157–174 (2014). https://doi.org/10.1007/s11786-014-0187-8

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  • DOI: https://doi.org/10.1007/s11786-014-0187-8

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