The Intermediate Value Theorem for Polynomials over Lattice‐ordered Rings of Functions

Abstract

ABSTRACTThe classical intermediate value theorem for polynomials with real coefficients is generalized to the case of polynomials with coefficients in a lattice‐ordered ring that is a subdirect product of totally ordered rings. Several candidates for a generalization are investigated, and particular attention is paid to the case when the lattice‐ordered ring is the algebra C(X) of continuous real‐valued functions on a completely regular topological space X. For all but one of these generalizations, the intermediate value theorem holds only if X is an F‐space in the sense of Gillman and Jerison. Surprisingly, for the most interesting of these generalizations, if X is compact, the intermediate value theorem holds only if X is an F‐space and each component of X is an hereditarily indecomposable continuum. It is not known if there is an infinite compact connected space in which this version of the intermediate value theorem holds.