Commutative algebras for arrangements

Orlik, Peter; Terao, Hiroaki

doi:10.1017/s0027763000004852

Cited by 60 publications

(73 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…• (Berget [4], Brion-Verge [5], Orlik-Terao [13], Proudfoot-Speyer [17], Terao [23]) Given a hyperplane arrangement determined by the linear functionals α 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Combinatorics and geometry of power ideals

Ardila¹,

Postnikov²

2010

Trans. Amer. Math. Soc.

148

View full text Add to dashboard Cite

Abstract. We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines.We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings.Our work unifies and generalizes results due to Dahmen-Micchelli, HoltzRon, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope.

show abstract

“…• (Berget [4], Brion-Verge [5], Orlik-Terao [13], Proudfoot-Speyer [17], Terao [23]) Given a hyperplane arrangement determined by the linear functionals α 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Combinatorics and geometry of power ideals

Ardila¹,

Postnikov²

2010

Trans. Amer. Math. Soc.

148

View full text Add to dashboard Cite

show abstract

“…As Remark 8.8. Various P-spaces without a dual D-space have been studied by several other authors e. g. [2,5,39,45,52]. The approach in [2,39] is slightly different from ours.…”

Section: Comparison With Previously Known Zonotopal Spacesmentioning

confidence: 84%

Zonotopal algebra and forward exchange matroids

Lenz

2016

Advances in Mathematics

View full text Add to dashboard Cite

Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X), P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms.The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known.The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.

show abstract

“…(0) (M) ab of the algebra 3T (0) (M) is isomorphic to the Orlik-Terao algebra [114], denoted by OT(M) (known also as even version of the Orlik-Solomon algebra, denoted by OS + (M) ) associated with matroid M [28]. Moreover, the anticommutative quotient of the odd version of the algebra 3T (0) (M), as we expect, is isomorphic to the Orlik-Solomon algebra OS(M) associated with matroid M, see, e.g., [11,49].…”

Section: Extended Abstractmentioning

confidence: 99%