Digit systems over commutative rings

2010/04/21 by Klaus Scheicher, Paul Surer, Scheicher, Klaus +5 · 1 citation
Mathematics · #11A63 #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.NT #msc:11A63

paper · pdf · doi:10.48550/arxiv.1004.3729

21 pages, 1 figure; submitted.

arxiv created 2010/04/21 · arxiv updated 2010/04/22

Abstract

Let \E be a commutative ring with identity and P∈\E[x] be a polynomial. In the present paper we consider digit representations in the residue class ring \E[x]/(P). In particular, we are interested in the question whether each A∈\E[x]/(P) can be represented modulo P in the form e₀+e₁ X + ⋯ + e_h X^h, where the e_i∈\E[x]/(P) are taken from a fixed finite set of digits. This general concept generalises both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

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