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

JOURNAL OF ALGEBRA

卷:502

Article

Cahen, Paul-Jean¹  Chabert, Jean-Luc² 

[1] 12 Traverse Lavoir Grand Mere, F-13100 Aix En Provence, France

[2] Univ Picardie, LAMFA, UMR CNRS 7352, 33 Rue St Leu, F-80039 Amiens, France

关键词: Integer-valued polynomials;    n-universal subsets;    Newton sequences;    Bhargava's factorial ideals;    Dedekind domains;   

DOI  :  10.1016/j.jalgebra.2018.01.020

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

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