Test sets for polynomials: n-universal subsets and Newton sequences

Abstract

Let E be a subset of an integral domain D with quotient field K. A subset S of E is said to be an n-universal subset of E if every integer-valued polynomial f(X)∈K[X] on S (that is, such that f(S)⊆D), with degree at most n, is integer-valued on E (that is, f(E)⊆D). A sequence a0,…,an of elements of E is said to be a Newton sequence of E of length n if, for each k≤n, the subset {a0,…,ak} is a k-universal subset of E. Our main results concern the case where D is a Dedekind domain, where both notions are strongly linked to p-orderings, as introduced by Bhargava. We extend and strengthen previous studies by Volkov, Petrov, Byszewski, Fra̧czyk, and Szumowicz that concerned only the case where E=D. In this case, but also if E is an ideal of D, or if E is the set of prime numbers >n+1 (in D=Z), we prove the existence of sequences in E of which n+2 consecutive terms always form an n-universal subset of E.