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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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