Restricted Elasticity and Rings of Integer-Valued Polynomials Determined by Finite Subsets

Let D be an integral domain such that Int(D) ≠ KX] where K is the quotient field of D. There is no known example of such a D so that Int(D) has finite elasticity. If E is a finite nonempty subset of D, then it is known that Int(E, D) = {f(X) ∈ KX] | f(e) ∈ D for all e ∈ E} is not atomic. In this note, we restrict the notion of elasticity so that it is applicable to nonatomic domains. For each real number r ≥ 1, we produce a ring of integer-valued polynomials with restricted elasticity r. We further show that if D is a unique factorization domain and E is finite with |E| > 1, then the restricted elasticity of Int(E, D) is infinite. Part of this work was completed while the first author was on an Academic Leave granted by the Trinity University Faculty Development Committee.