Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings

We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such generalized chromatic polynomials are determined by $h$-vectors of auxiliary simplicial complexes. In addition to generalizing related results on graphs and matroids, the simplicial complexes used allow us to consider problems that are not necessarily analogues of those considered for graphs. Some examples include supports of cyclotomic polynomials, log concavity properties of a polynomial or some translate of the polynomial, and symmetry relations between a polynomial and its reciprocal polynomial. If the $h$-vectors involed have sufficiently large entries, the Hilbert series are Hilbert polynomials of some $k$-algebra. As a consequence of connections between $h$-vectors and simplicial chromatic polynomials, we also find simplicial complexes whose $h$-vectors are determined by addition-contraction relations of simplicial complexes analogous to deletion-contraction relations of graphs. The constructions used involve generalizations of relations Euler characteristics of configuration spaces and chromatic polynomials of graphs.