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Equivariant Chow cohomology of nonsimplicial toric varieties
 Transactions of the A.M.S
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LatticeSupported Splines on Polytopal Complexes
 IN APPL. MATH
, 2014
"... We study the module Cr(P) of piecewise polynomial functions of smoothness r on a pure ndimensional polytopal complex P ⊂ Rn, via an analysis of certain subcomplexes PW obtained from the intersection lattice of the interior codimension one faces of P. We obtain two main results: first, we show tha ..."
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We study the module Cr(P) of piecewise polynomial functions of smoothness r on a pure ndimensional polytopal complex P ⊂ Rn, via an analysis of certain subcomplexes PW obtained from the intersection lattice of the interior codimension one faces of P. We obtain two main results: first, we show that the vector space Crd(P) of splines of degree ≤ d has a basis consisting of splines supported on the PW for d 0. We call such splines latticesupported. This shows that an analog of the notion of a starsupported basis for Crd(∆) studied by AlfeldSchumaker in the simplicial case holds [3]. Second, we provide a pair of conjectures, one involving latticesupported splines, bounding how large d must be so that dimRCrd(P) agrees with the McDonaldSchenck formula [14]. A family of examples shows that the latter conjecture is tight. The proposed bounds generalize known and conjectured bounds in the simplicial case.
ALGEBRAIC METHODS IN APPROXIMATION THEORY
"... This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhed ..."
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This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in Rk, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a handson introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree. A tutorial on computational
ASSOCIATED PRIMES OF SPLINE COMPLEXES
, 2014
"... The spline complex R/J [Σ] whose top homology is the algebra Cα(Σ) of mixed splines over the fan Σ ⊂ Rn+1 was introduced by SchenckStillman in [26] as a variant of a complex R/I[Σ] of Billera [5]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we ..."
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The spline complex R/J [Σ] whose top homology is the algebra Cα(Σ) of mixed splines over the fan Σ ⊂ Rn+1 was introduced by SchenckStillman in [26] as a variant of a complex R/I[Σ] of Billera [5]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of Cα(Σ), including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of Σ, generalizing the computations in [14, 19]. The second is a description of the fourth coefficient of the Hilbert polynomial of HP (Cα(Σ)) for simplicial fans Σ. We use this to derive the result of Alfeld, Schumaker, and Whiteley on the generic dimension of C1 tetrahedral splines for d 0 [3] and indicate via an example how this description may be used to give the fourth coefficient in particular nongeneric configurations.