Abstract
To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration.
Highlights
Polynomial ideals generated by powers of linear forms, often called power ideals, appear in a number of mathematical contextsThis paper is concerned with a family of power ideals associated to a vector configuration
These were originally introduced in the context of multivariate approximation theory, mainly as a tool to study the space spanned by the local polynomial pieces of a box spline and their derivatives [11]
Such power ideals are known to strongly reflect combinatorial aspects of the underlying vector configuration, and have generated renewed interest in recent years, owing to their rich geometry and combinatorics and to their relevance in subjects as varied as the cohomology of homogeneous manifolds and Cox rings We delay a precise definition until Section 2
Summary
Polynomial ideals generated by powers of linear forms, often called power ideals, appear in a number of mathematical contexts (see [3] and the references therein.). The exponent vectors of the standard monomials modulo such a (monomial) power ideal can be readily identified with the lattice points of a polymatroid (a.k.a. generalized permutahedron), a convex polytope defined by a submodular function (Corollary 4.8) We illustrate both results in Example 1.1 below. The power ideal associated to a vector configuration V defines a graded ring whose Hilbert function is known to coincide with the h-vector of the (abstract) simplicial complex of subsets T of V such that span(V T ) = span(V ) (cf Remark 5.5) We postpone the proof of Lemma 2.3 to Section 5, to highlight the relation between power ideals and Stanley–Reisner theory
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