Combinatorics and geometry of power ideals

Abstract

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.

Interpolation, Box Splines, and Lattice Points in Zonotopes

“…Exactness: Exactness of the first row was stated in [2] and proven in [20]. The proof relies on the fact that P − (X) can be written as the kernel of a power ideal.…”

“…The space P(X) first appeared in approximation theory [1,10,16]. Later, spaces of this type and generalisations were also studied by authors in other fields, e. g. [2,4,19,20,21,24]. We will let the elements of P − (X) act as differential operators on the box spline.…”

“…This is a consequence of a deep connection between the spaces P − (X) and P(X) and matroid theory. The Hilbert series of these two spaces are evaluations of the Tutte polynomial of the matroid defined by X [2]. One can deduce that the Hilbert series of the internal space is equal to the h-polynomial of the broken-circuit complex [5] of the matroid M * (X) that is dual to the matroid defined by X and the Hilbert series of the central space equals the h-polynomial of the matroid complex of M * (X).…”

Zonotopal algebra and forward exchange matroids

“…It is known that there is a canonical way to construct bases for the spaces of P-type [2,27,34,40,42]. The first of the two main results in this paper is an algorithm that constructs a basis for spaces of D-type.…”

“…Federico Ardila and Alex Postnikov studied generalised P-spaces and connections with power ideals [2]. Bernd Sturmfels and Zhiqiang Xu established a connection with Cox rings [51].…”