Open Access Publications from the University of California

The Generalized External Order, and Applications to Zonotopal Algebra

• Author(s): Gillespie, Bryan Rae
We then apply this theory to improve our understanding of certain constructions in zonotopal algebra. We first explain the fundamental link between zonotopal algebra and the external order by characterizing Lenz's forward exchange matroids in terms of the external order. Next we describe the behavior of Lenz's zonotopal $\mathcal{D}$-basis polynomials under taking directional derivatives, and we use this understanding to provide a new algebraic construction for these polynomials. The construction in particular provides the first known algorithm for computing these polynomials which is computationally tractible for inputs of moderate size. Finally, we provide an explicit construction for the zonotopal $\mathcal{P}$-basis polynomials for the internal and semi-internal settings.