Infinitely integer-valued polynomials over an algebraic number field

K. Rogers and Ernst Gabor Straus

Vol. 118 (1985), No. 2, 507–522

DOI: 10.2140/pjm.1985.118.507

Abstract

Let K be an algebraic number field and R its ring of integers. A polynomial f over K is integer-valued iff f(R) ⊆ R: it is infinitely integer-valued, written f ∈ D^∞(R), iff f and all its derivatives are integer-valued. For each K we construct a sequence of ideals A_k of R, and a sequence of polynomials H_k(x) over R, such that a polynomial f of degree n lies in D^∞(R) if and only if it is of the form a₀H₀(x)∕0! + ⋯ + a_nH_n(x)∕n!, with a_k in A_k, k = 0,1,…,n.

Mathematical Subject Classification 2000

Primary: 11R09

Milestones

Received: 31 May 1984

Published: 1 June 1985