Newtonian and Schinzel sequences in a domain

Abstract

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 , { u s q , … , u ( s + 1 ) q − 1 } 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. Our 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. We show 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 . All results are refined to sequences of finite length.