Newtonian and Schinzel sequences in a domain

A Schinzel or FF sequence in a domain is such that, for every ideal II with norm qq, its first qq terms form a system of representatives modulo II, and a Newton or NN sequence such that the first qq terms serve as a test set for integer-valued polynomials of degree less than qq. Strong FF and strong NN sequences are such that one can use any set of qq consecutive terms, not only the first ones, finally a very well FF ordered sequence, for short, a V.W.F   sequence, is such that, for each ideal II with norm qq, and each integer s,{usq,…,u(s+1)q−1}s,{usq,…,u(s+1)q−1} is a complete set of representatives modulo II. In a quasilocal domain, V.W.F   sequences and NN sequences are the same, so are strong FF and strong NN sequences. Our main result is that a strong NN sequence is a sequence which is locally a strong FF sequence, and an NN sequence a sequence which is locally a V.W.F  . sequence. We show that, for FF sequences there is a bound on the number of ideals of a given norm. In particular, a sequence is a strong FF sequence if and only if it is a strong NN sequence and for each prime pp, there is at most one prime ideal with finite residue field of characteristic pp. All results are refined to sequences of finite length.