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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) 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.
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https://hal-u-picardie.archives-ouvertes.fr/hal-03621211
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Soumis le : lundi 28 mars 2022 - 09:37:47
Dernière modification le : mardi 29 mars 2022 - 03:58:28

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Paul-Jean Cahen, Jean-Luc Chabert. Test sets for polynomials: n-universal subsets and Newton sequences. Journal of Algebra, 2018, 502, pp.277-314. ⟨10.1016/j.jalgebra.2018.01.020⟩. ⟨hal-03621211⟩

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