The Intermediate Value Theorem for f-Rings
by
Suzanne Larson
Loyola Marymount University

The classical Intermediate Value Theorem (IVT) states that if f is a continuous function mapping the interval [a, b] into R, with f(a) <= 0, and f(b) >= 0, then there is a c in [a, b] such that f(c) = 0. Three forms of the IVT restricted to polynomial functions have been investigated in the context of f-rings by Henriksen, Larson and Martinez, and in C(X) by Dow and Hart. The specific definition of one of these forms follows. Suppose A is a commutative f -ring with identity element, A[t] denote the ring of polynomials with coefficients in A and let B subset or equal A[t]. If, for every p(t) in B, and distinct elements u, v in A such that p(u) > 0 and p(v) < 0, there is a w in A such that p(w) = 0 and u /\ v <= w <= u \/ v, we say the intermediate value theorem holds for B in A. If B = A[t], we say that A is an IVT f-ring. We will discuss new results concerning properties of f-rings possessing one of the three forms of the IVT property. In addition, we will describe characterizations of those f-rings that are weak IVT, IVT or strong IVT for various classes of polynomials. Finally, we will discuss open questions relating to these f-rings.

Date received: February 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caed-03.