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Non-unique factorizations in orders of global fields. (English) Zbl 0812.11061
Let \(O\) be an order in an algebraic field \(K\). The authors obtain asymptotics for the number of principal ideals \(I\subset O\) with \([O:I] \leq x\), the generator of which either has only factorizations of length not exceeding \(k\) or has factorizations of at most \(k\) lengths or has at most \(k\) distinct factorizations. In all three cases this number turns out to be asymptotically equal to \(Cx(\log x)^{-a} (\log\log x)^ b\) with suitable \(C>0\), \(0\leq a\leq 1\) and a nonnegative integer \(b\), having combinatorial interpretation. Here \(C\) depends on \(k\) and \(O\) but \(a\), \(b\) depend only on \(k\) and the structure of the Picard group of \(O\) and in the first case one has \(a=1\). The obtained results are similar to those obtained earlier in the case of principal orders [A. Geroldinger, Acta Arith. 57, 365-373 (1991; Zbl 0729.11054); F. Halter-Koch, ibid. 62, 173-206 (1992; Zbl 0762.11041); F. Halter- Koch and W. Mueller, J. Reine Angew. Math. 421, 159-188 (1991; Zbl 0736.11064); J. Kaczorowski, Acta Arith. 43, 53-68 (1983; Zbl 0526.12006)]. The authors utilize an axiomatic approach, which allows them to obtain at the same time similar assertions for more general structures, including generalized Hilbert semigroups.

11R27 Units and factorization
11N45 Asymptotic results on counting functions for algebraic and topological structures
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20M14 Commutative semigroups
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