Combinatorics and geometry of power ideals

TL;DR

This paper explores the algebraic and geometric properties of power ideals generated by powers of linear forms, linking their Hilbert series to combinatorial invariants of hyperplane arrangements and unifying various prior results.

Contribution

It establishes that the Hilbert series of certain power ideals are determined by the combinatorics of hyperplane arrangements via the Tutte polynomial, generalizing previous work.

Findings

Hilbert series are determined by the Tutte polynomial of arrangements

Formulas for Hilbert series of fat point ideals and Cox rings are derived

Settles a conjecture on spline interpolation on zonotopes

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 the resulting 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 interior lattice points of a zonotope.