Interpolation, Box Splines, and Lattice Points in Zonotopes

Abstract

Abstract. Let X be a totally unimodular list of vectors in some lattice. Let B X be the box spline defined by X. Its support is the zonotope Z(X). We show that any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form p(D)B X in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.

Internal zonotopal algebras and the monomial reflection groups G(m,1,n)

“…In the external and central cases, bases of the Macaulay inverse system of the associated ideals are given by certain products of linear forms defining A. Analogous results were conjectured to hold in the internal case and an incorrect proof was given in [1]. This was corrected in [2], and a combinatorial basis was later given by Lenz [15] when A is unimodular. For non-unimodular A, there is no known canonical basis of the Macaulay inverse system of I A,−2 described by the matroid of A.…”

“…Of recent interest are the so-called zonotopal algebras of a hyperplane arrangement [1,2,12,14,15]. The zonotopal ideals of an arrangement A ⊂ V are ideals in Sym(V ) generated by powers of elements of v. Specifically, the kth zonotopal ideal of A is…”

“…The internal case k = −2 exhibits dramatic subtleties in comparison to all other cases k > −1 [2], [14,Section 7.4]. The initial motivation for studying the internal zonotopal ideal was that its Macaulay inverse system consists of those partial differential operators that leave continuous the box spline associated to A (assuming A is unimodular) [15,Corollary 9].…”

“…Of recent interest are the so-called zonotopal algebras of a hyperplane arrangement [1,2,12,14,15]. The zonotopal ideals of an arrangement A ⊂ V are ideals in Sym(V ) generated by powers of elements of v. Specifically, the kth zonotopal ideal of A is I A,k = h max{ρ A (h)+k+1,0} : h ∈ V where ρ A (h) is the number of hyperplanes in A that do not contain h. The quotient S A,k = Sym(V )/I A,k is the kth zonotopal algebra of A. Holtz and Ron [12] single out the cases k = −2, −1, 0 as being of particular interest, and call these the internal, central and external zonotopal algebras of A.…”

Two counterexamples for power ideals of arrangements

“…The arrangement above is unimodular, but it does not provide a counterexample to [3, Conjecture 1.8]. In fact, Matthias Lenz [4] has recently put forward a proof of this weaker conjecture.…”