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An analysis using the Zaks-Skula constant of element factorizations in Dedekind domains. (English) Zbl 0784.11050
For a Dedekind domain \(D\) let \(\rho(D)\) be the least upper bound for the ratio of the lengths of two factorizations into irreducibles of a non- unit element of \(D\). This function has been studied by J. Steffan [J. Algebra 102, 229-236 (1986; Zbl 0593.13015)] and R. J. Valenza [J. Number Theory 36, 212-218 (1990; Zbl 0721.11043)] and called the elasticity of \(D\). Further let \(\Phi(n)\) be the number of distinct lengths of factorizations which a product of \(n\) irreducible elements in \(D\) may have. An upper bound for \(\rho(D)\) is established which permits the study of the asymptotic behaviour of \(\Phi(n)\). In a later paper [J. Number Theory 43, 24-30 (1993; Zbl 0765.11043)] the authors showed that if \(D\) has a finite class group \(G\) and there are prime ideals in every ideal class then one has \(\lim \Phi(n)/n= (D(G)^ 2-4)/2D(G)\), where \(D(G)\) denotes the Davenport’s constant of \(G\).

MSC:
11R27 Units and factorization
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20D06 Simple groups: alternating groups and groups of Lie type
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