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Bounding the Roots of Ideal and Open Set Polynomials

Authors:	Jason I. Brown Carl A. Hickman Hugh Thomas David G. Wagner

Affiliation:	(1) Department of Mathematics and Statistics and Faculty of Computer Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada;(2) The Fields Institute for Research in Mathematical Sciences, University of Toronto, Toronto, Ontario, M5T 3J1, Canada;(3) Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada;(4) Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Abstract:	Let P be a preorder (i.e., reflexive, transitive relation) on a finite set X. The ideal polynomial of P is the function ${text{ideal}}_{P} (x) = {sumnolimits_{k geqslant 0} {d_{k} x^{k} ,} }$ 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. Received December 15, 2004

Keywords:	05A15 06A11 12D10 54A10
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