The Intermediate Value Theorem for Polynomials over Lattice-ordered Rings of Functions
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- text
- intermediate value theorem
- IVT ring
- IVT space
- strong IVT ring
- strong IVT space
- f-ring
- semiprime f-ring
- 1-convex f-ring
- maximal ideal
- prime ideal
- minimal prime ideal
- F-space
- zero-dimensional
- strongly zero-dimensional
- compact space
- Lindelöf space
- Stone-Čech compactification
- connected space
- component
- continuum
- indecomposable continuum
- hereditarily indecomposable continuum
- valuation domain
- SV-space
- Mathematics
- Physical Sciences and Mathematics