Newtonian and Schinzel sequences in a domain
type: Item
instance of: scholarly article
title: Newtonian and Schinzel sequences in a domain
zbMATH Open document ID: 1170.13007
author: Paul-Jean Cahen
published in: Journal of Pure and Applied Algebra
publication date: 2009-07-02T00:00:00Z
review text: A Schinzel or \(F\) sequence in a domain is such that, for every ideal \(I\) with norm \(q\), its first \(q\) terms form a system of representatives modulo \(I\), and a Newton or \(N\) sequence such that the first \(q\) terms serve as a test set for integer-valued polynomials of degree less than \(q\). Strong \(F\) and strong \(N\) sequences are such that one can use any set of \(q\) consecutive terms, not only the first ones, finally a very well \(F\) ordered sequence, for short, a \(V.W.F\) sequence, is such that, for each ideal \(I\) with norm \(q\), and each integer \(s\),\(\left\{u_{sq},\cdots,u_{(s+1)q-1}\right\}\) is a complete set of representatives modulo \(I\). In a quasilocal domain, \(V.W.F\) sequences and \(N\) sequences are the same, so are strong \(F\) and strong \(N\) sequences. The author of this paper proved many results in this concern, among them: A Schinzel domain is a principal ideal domain in finite norms and that a Newtonian domain is a Dedekind domain in finite norms, where these properties are defined by considering only the ideals with finite norm. The main result is that a strong \(N\) sequence is a sequence which is locally a strong \(F\) sequence, and an \(N\) sequence a sequence which is locally a \(V.W.F.\) sequence. The author exhibits examples showing that non of these implications can be reversed. He shows that, for \(F\) sequences there is a bound on the number of ideals of a given norm. In particular, a sequence is a strong \(F\) sequence if and only if it is a strong \(N\) sequence and for each prime \(p\), there is at most one prime ideal with finite residue field of characteristic \(p\). In the case of \(F\) sequences he showed that there is a limit on the number of ideals with small norm. It follows that in a Schinzel domain, there cannot be more than \(q\) ideals with norm \(q\), moreover, if there are ideals both of norm 2 and 3, there cannot be more than one ideal of each kind. All results of the paper are, also refined to…
reviewed by: Jebrel M. Habeb
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 5574532
zbMATH Keywords: Schinzel sequence
zbMATH Keywords: Newtonian sequence
zbMATH Keywords: Dedekind domain
zbMATH Keywords: norm
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.03.004
OpenAlex ID: W1965140997
cites work: Q5620634
cites work: Q4115215
cites work: On Sequences of Algebraic Integers
cites work: Remark on a problem of Schinzel
cites work: P-orderings and polynomial functions on arbitrary subsets of Dedekind rings.
cites work: Generalized factorials and fixed divisors over subsets of a Dedekind domain
cites work: The Factorial Function and Generalizations
cites work: Anneaux de Bhargava
cites work: Q5541476
cites work: Q3436759
cites work: \(P\)-orderings: A metric viewpoint and the non-existence of simultaneous orderings.
cites work: On sequences of algebraic integers in pure extensions of prime degree
cites work: Integer-valued polynomials
cites work: Q5521594
cites work: Simultaneous orderings in function fields
DOI: 10.1016/J.JPAA.2009.03.004
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description: scientific article; zbMATH DE number 5574532