Graphs, Syzygies, and Multivariate Splines

Rose, Lauren L.

doi:10.1007/s00454-004-1141-3

Cited by 20 publications

(23 citation statements)

References 16 publications

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“…From Lemma 4.10 and Example 4.11 we see that it is useful to understand the homology of the complexes R[Σ W,σ , Σ W,σ ]. To this end we introduce a variant of a graph used by Schenck [28, Definition 2.5], which also builds on dual graphs of Rose [24,25]. This graph simplifies the computation of the homology of Σ W,σ in homological degree dim(W ) + 1.…”

Section: Figurementioning

confidence: 99%

Associated primes of spline complexes

DiPasquale

2016

Journal of Symbolic Computation

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Abstract. The spline complex R/J [Σ] whose top homology is the algebra C α (Σ) of mixed splines over the fan Σ ⊂ R n+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 C α (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 (C α (Σ)) for simplicial fans Σ. We use this to derive the result of Alfeld, Schumaker, and Whiteley on the generic dimension of C 1 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.

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Section: Figurementioning

confidence: 99%

Associated primes of spline complexes

DiPasquale

2016

Journal of Symbolic Computation

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“…This method has been refined and extended by Schenck,Stillman,and McDonald ([20] and [14]). The last of these gives a polyhedral version of the Alfeld-Schumaker formula in the planar case, building on work of Rose [17], [18] on dual graphs.…”

Section: Introductionmentioning

confidence: 99%

Lattice-supported splines on polytopal complexes

DiPasquale

2014

Advances in Applied Mathematics

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Abstract. We study the module C r (P) of piecewise polynomial functions of smoothness r on a pure n-dimensional polytopal complex P ⊂ R n , via an analysis of certain subcomplexes P W 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 C r d (P) of splines of degree ≤ d has a basis consisting of splines supported on the P W for d 0. We call such splines lattice-supported. This shows that an analog of the notion of a star-supported basis for C r d (∆) studied by Alfeld-Schumaker in the simplicial case holds [3]. Second, we provide a pair of conjectures, one involving lattice-supported splines, bounding how large d must be so that dim R C r d (P) agrees with the McDonald-Schenck 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.

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“…As noted in Example 2.3, a syzygy on the linear forms adjacent to a single fixed vertex corresponds to a loop around that vertex. In [12], Rose shows that if the dual graph G P of P has a basis of disjoint cycles, then the projective dimension and Hilbert series of C r (P ) are determined by the case when G P has a single cycle, which Rose analyzed in [11]. We now define a refined version of dual graph, which depends on P and the choice of a codimension-2 linear subspace.…”

Section: 2mentioning

confidence: 99%

“…It is possible to consider all the C r k (P ) at once by passing to a module over a polynomial ring, and in [20] Yuzvinsky uses sheaves on posets to obtain results on the freeness of this module. In [11], [12], Rose studies cycles on the dual graph of P . Our key technical innovation is a refined version of the dual graph, depending on a choice of codimension-2 linear space; used in conjunction with localization techniques.…”

Section: Introductionmentioning

confidence: 99%

Piecewise polynomials on polyhedral complexes

McDonald¹,

Schenck²

2009

Advances in Applied Mathematics

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For a d-dimensional polyhedral complex P , the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f (P, r, k) of degree d. When d = 2 and P is simplicial, in [1] Alfeld and Schumaker give a formula for all three coefficients of f . However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension-2 linear spaces, we obtain the first three coefficients of f (P, r, k), giving a complete answer when d = 2.2000 Mathematics Subject Classification. Primary 41A15, Secondary 13D40, 52C99.

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Graphs, Syzygies, and Multivariate Splines

Cited by 20 publications

References 16 publications

Associated primes of spline complexes

Associated primes of spline complexes

Lattice-supported splines on polytopal complexes

Piecewise polynomials on polyhedral complexes

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