A Family of Ideals of Minimal Regularity and the Hilbert Series ofC(Δ)

Schenck, Hal; Stillman, Michael

doi:10.1006/aama.1997.0533

Cited by 34 publications

(51 citation statements)

References 9 publications

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“…The following results are generalizations of Theorem 5.3(a) in [13] and Theorem 5.2 in [12], respectively.…”

Section: Lemma 412 If E 1 Is a Removable Edge Of C Then It Is An Rmentioning

confidence: 75%

“…If we can recursively split off all of the cycles of G, Schenck and Stillman assert in [12] that this is the best possible result using only local data.…”

Section: Theorem 415 If Every Interior Edge Is a Removable Edge Of mentioning

confidence: 99%

“…Using the terminology in [13], we define a non-boundary edge of to be a pseudoboundary edge if its affine span is the same as that of an edge meeting the outside boundary, and both edges share a cycle in G. The following theorem is a extension of Theorem 5.3(b) in [12] to polygonal complexes. Proof.…”

Section: Example 52mentioning

confidence: 99%

“…When B r ( ) is graded this decomposition also gives a straightforward way of computing its Hilbert Series. We describe conditions on G, L, and r for such a decomposition to occur, providing alternate proofs for some results of Schenck and Stillman from [12] and [13], and extending those results to polyhedral complexes. Schenk and Stillman claim that these are the best possible results using only local data.…”

Section: Introductionmentioning

confidence: 96%

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Graphs, Syzygies, and Multivariate Splines

Rose

2004

Discrete Comput Geom

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The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension and the Hilbert series of the module. We provide alternate proofs of some results of Schenck and Stillman, extending those results to the polyhedral case. We also provide examples which illustrate the role that geometry plays in determining the syzygy module.

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“…The following results are generalizations of Theorem 5.3(a) in [13] and Theorem 5.2 in [12], respectively.…”

Section: Lemma 412 If E 1 Is a Removable Edge Of C Then It Is An Rmentioning

confidence: 75%

“…If we can recursively split off all of the cycles of G, Schenck and Stillman assert in [12] that this is the best possible result using only local data.…”

Section: Theorem 415 If Every Interior Edge Is a Removable Edge Of mentioning

confidence: 99%

Section: Example 52mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Graphs, Syzygies, and Multivariate Splines

Rose

2004

Discrete Comput Geom

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“…Next, let denote the 12-split triangulation of a single triangle . For future reference, we state the following formula for the dimension of the space of C r splines of degree d on , which is a special case of Theorem 3.1 in [2] (also compare [23]). …”

Section: Dimension Formulasmentioning

confidence: 99%

Stable Simplex Spline Bases for $$C^3$$ C 3 Quintics on the Powell–Sabin 12-Split

Lyche

Muntingh

2016

Constr Approx

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Abstract. For the space of C 3 quintics on the Powell-Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the L∞ norm with a condition number independent of the geometry, have a well-conditioned Lagrange interpolant at the domain points, and a quasi-interpolant with local approximation order 6. We show an h 2 bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases we derive C 0 , C 1 , C 2 and C 3 conditions on the control points of two splines on adjacent macrotriangles.

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Fat Points, Inverse Systems, and Piecewise Polynomial Functions

Geramita¹,

Schenck²

1998

Journal of Algebra

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ŽWe explore the connection between ideals of fat points which correspond to n Ž . . subschemes of ‫ސ‬ obtained by intersecting mixed powers of ideals of points , and Ž . piecewise polynomial functions splines on a d-dimensional simplicial complex ⌬ d w x embedded in R . Using the inverse system approach introduced by Macaulay 11 , we give a complete characterization of the free resolutions possible for ideals in w x Ž k x, y generated by powers of homogeneous linear forms we allow the powers to . differ . We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed ⌬, the dimension of the vector space of splines on ⌬ of degree less than or equal to k. We use this relationship and the results above to derive a formula which gives the number of Ž . planar mixed splines in sufficiently high degree. ᮊ

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A Family of Ideals of Minimal Regularity and the Hilbert Series ofC(Δ)

Cited by 34 publications

References 9 publications

Graphs, Syzygies, and Multivariate Splines

Graphs, Syzygies, and Multivariate Splines

Stable Simplex Spline Bases for $$C^3$$ C 3 Quintics on the Powell–Sabin 12-Split

Fat Points, Inverse Systems, and Piecewise Polynomial Functions

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