Truncated power splines: acyclic resolutions of cone polynomials

Cone polynomials in n > 1 variables, also known as volume polynomials and/or spline polynomials, are the polynomials that appear in the local structure of the truncated powers, hence in the local structure of any derived construction such as box splines, simplex splines, partition functions, character formulas and moment maps. The underlying geometry is determined by a real linear matroid, i.e., a real matrix X n × N of rank n. The polynomial space itself is defined as the kernel D(X) of an ideal J (X) of differential operators, whose generators, each, are products of linear forms. While important statistics on D(X) (e.g., its Hilbert series) are classically known, its algebraic structure is considered to be hopelessly involved. In particular, as of today, and save a handful of truly rudimentary cases, basis constructions for D(X) are scarce, and provide, perhaps, neither an insight into the polynomials that make D(X), nor an aid in pertinent applications. We study the above setup when X is the incidence matrix of a graph G, and focus only on the socle soc(D(X)) of D(X), which is comprised of the top-degree homogeneous polynomials in D(X): the polynomial pieces that make the truncated powers span that socle only. We first resolve the ideal J (X) by representing it as the intersection of larger ideals, each of which a much simpler one: a complete intersection (CI) ideal. Each CI ideal JGι is induced by an acyclic directed version Gι of the graph G. Kernels of CI ideals have 1-dimensional socles, and the final outcome is a resolution of soc(D(X)) into a direct sum of these 1-dimensional socles: (0.1) soc(D(X)) = ⊕Gιsoc(JGι⊥). “lababs This decomposition can be thought of as an algebraic realization of a known combinatorial graph identity, i.e., that the number of spanning trees of the graph with 0 external activity (which is known to be equal of dim soc(D(X))) is the same as the number of acyclic orientations of G with one fixed source. We then provide an explicit combinatorial algorithm for the construction of the 1dimensional socles of JGι⊥. This explicit construction leads to the following core, surprising, observation: when writing each polynomial in the basis provided in (0.1) as a combination of monomials, the monomial coefficients are determined by a discrete truncated power (i.e., a partition function) in dimension n − 1. That means that not only truncated powers in n dimensions are piecewise in the polynomial space soc(D(X)), but also, in a suitable sense, this latter polynomial space is canonically isomorphic to a suitable discrete truncated power space of a lower dimension. In short, cone polynomials underlie the structure of truncated powers, while truncated powers underlie the structure of cone polynomials!