P-orderings and polynomial functions on arbitrary subsets of Dedekind rings

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Abstract

We introduce the notion of a P-ordering of an arbitrary subset X of a Dedekind ring R, and use it to investigate the functions from X to R which can be represented by polynomials. In the case when R is a finite principal ideal ring, our results include canonical representations for those polynomials which vanish on X, a canonical representation for each polynomial function from X to R, necessary and sufficient congruence conditions for a function from X to R to be a polynomial function, and a formula for the number of such functions. When R is a Dedekind domain, we use the concept of P-ordering to give necessary and sufficient conditions for the existence of a regular basis for the ring Int(X, R), and moreover, we give an explicit construction of such a basis whenever it exists. Finally, we deduce many of the classical theorems of Kempner, Carlitz, Pólya, and others on polynomial mappings as special cases of our results.

Original languageEnglish (US)
Pages (from-to)101-127
Number of pages27
JournalJournal fur die Reine und Angewandte Mathematik
Volume490
StatePublished - Dec 1 1997

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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