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

## Document Type

Article

## Department

Mathematics (HMC)

## Publication Date

1996

## Abstract

The 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.

## Rights Information

© 1996 New York Academy of Sciences, published by Wiley

## DOI

10.1111/j.1749-6632.1996.tb36802.x

## Recommended Citation

Henriksen, Melvin Separate vs. joint continuity. A tale of four topologies–-a summary. Proceedings of the Tennessee Topology Conference (Nashville, TN, 1996), 67–84, World Sci. Publishing, River Edge, NJ, 1997. DOI: 10.1111/j.1749-6632.1996.tb36802.x