Combinatorics and geometry of power ideals

By Federico Ardila, Alexander Postnikov. 2008

74 citations recorded by the source corpus.

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, Holtz-Ron, 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e87570117e00d4b36f0dd9b078ecf14c9f1ace32

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In Algebra and Number Theory

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Citations in the corpus

On the Weak Lefschetz Property for Powers of Linear Forms (2010, methodology): “For example, their Hilbert functions have been the focus of the papers [Ardila and Postnikov 2010; Sturmfels and Xu 2010; Harbourne et al. 2011], among others.”
Arithmetic matroids, the Tutte polynomial and toric arrangements (2011, background): “box splines in numerical analysis, root systems and parking functions in combinatorics are only some of the most well-known examples. More recently, Holtz and Ron in [16], and Ardila and Postnikov in [1] introduced various algebraic structures capturing a rich description of these objects. A central”
Algebraic and Geometric Methods in Enumerative Combinatorics (2014, background): “X ainNn T M(a)(x;y) (x 1)r(supp(a)) w a1 1 Qw n n = 1 i=1 (1 w i) Ze M (x 1)(y 1); (y 1)w 1 1 yw 1 ;:::; (y 1)w n 1 yw n For an algebraic interpretation of the multivariate Tutte polynomial, see [AP10]. The arithmetic Tutte polynomial When we have a collection AZnof integer vectors, there is a var”
A Tutte polynomial for toric arrangements (2009, background): “The relations between zonotopes and Dahmen-Micchelli spaces has been studied intensively in recent years, giving rise to many algebraic and combinatorial constructions (see in particular, [9], [13], [10], [2], [14], [18]).”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “Besides the connection with fat points and secant varieties, they are related to several areas of algebra, geometry and combinatorics; see [56].”