Test sets for polynomials: n-universal subsets and Newton sequences
Résumé
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) is an element of K[X] on S (that is, such that f(S) subset of D), with degree at most n, is integer valued on E (that is, f(E) subset of D). A sequence a(0),...,a(n) of elements of E is said to be a Newton sequence of E of length n if, for each k <= n, the subset \a(0),...,a(k)\ 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, Fraczyk, 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. (C) 2018 Elsevier Inc. All rights reserved.