We extend the definition of coarse flag Hilbert–Poincaré series to matroids; these arise in context local Igusa zeta functions associated hyperplane arrangements. study case oriented matroids by applying geometric and combinatorial tools related their topes. In this case, we prove that numerators are coefficient-wise bounded below Eulerian polynomial equality holds if only all topes simplicial....

We introduce a method for associating chain complex to module over combinatorial category, such that if the is exact then has rational Hilbert series. prove homology--vanishing theorems these complexes several categories including: category of finite sets and injections, opposite surjections, dimensional vector spaces field injections. Our main applications are modules known as $FS^{op}$ module...

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ ann −s−ū, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hil...