2021/03/05 by Joshua Maglione, Maglione, Joshua, Christopher Voll +1
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Alkaloids: synthesis and pharmacology
paper · pdf · doi:10.48550/arxiv.2103.03640
We introduce and study a class of multivariate rational functions associated\nwith hyperplane arrangements, called flag Hilbert-Poincar 'e series. These\nseries are intimately connected with Igusa local zeta functions of products of\nlinear polynomials, and their motivic and topological relatives. Our main\nresults include a self-reciprocity result for central arrangements defined over\nfields of characteristic zero. We also prove combinatorial formulae for a\nspecialization of the flag Hilbert-Poincar 'e series for irreducible Coxeter\narrangements of types \A, \B, and \D in terms of\ntotal partitions of the respective types. We show that a different\nspecialization of the flag Hilbert-Poincar 'e series, which we call the coarse\nflag Hilbert-Poincar 'e series, exhibits intriguing nonnegativity features and\n- in the case of Coxeter arrangements - connections with Eulerian polynomials.\nFor numerous classes and examples of hyperplane arrangements, we determine\ntheir (coarse) flag Hilbert-Poincar 'e series. Some computations were aided by\na SageMath package we developed.\n