Newtonian and Schinzel sequences in a domain
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 sequences of finite length.
scientific article; zbMATH DE number 5574532
scientific article