Bounding the Roots of Ideal and Open Set Polynomials

Abstract.

Let P be a preorder (i.e., reflexive, transitive relation) on a finite set X. The ideal polynomial of P is the function ${\text{ideal}}_{P} (x) = {\sum\nolimits_{k \geqslant 0} {d_{k} x^{k} ,} }$ where d _k is the number of ideals (i.e. downwards closed sets) of cardinality k in P. We provide upper bounds for the moduli of the roots of ideal _P (x) in terms of the width of P. We also provide examples of preorders with roots of large moduli. The results have direct applications to the generating polynomials counting open sets in finite topologies.