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Quantitative aspects of non-unique factorization: A general theory with applications to algebraic function fields. (English) Zbl 0736.11064
From the authors’ introduction: “Let \(R\) be a Dedekind domain; then every non-unit \(0\neq\alpha\in R\) has a factorization and the ideal class group \(G\) of \(R\) is said to measure the deviation of \(R\) from unique factorization . \((\dots)\) If \(\| \|\) is a reasonable size function on \(R\), one may ask for the asymptotic behaviour of the number of elements in certain subsets of \(R\) defined by factorization properties. Among others, the following sets have been studied.
\(M_ k\): the set of all \(\alpha\in R\) having a factorization of length \(k\),
\(F_ k\): the set of all \(\alpha\in R\) having at most \(k\) distinct factorizations,
\(G_ k\): the set of all \(\alpha\in R\) having factorization of at most \(k\) different lengths.
For any subset \(Z\subset R\) and \(x\geq1\) let \(Z(x)\) be the number of elements \(\alpha \in Z\) satisfying \(\|\alpha\|\leq x\), where from each set of associated elements only one is counted (such that, in fact, only the principal ideals \(\alpha R\) are counted). If \(R\) is the ring of integers of an algebraic number field, then \(\|\alpha\|=(R:\alpha R)\) is a reasonable size function, and the following results concerning \(M_ 1(x)\), \(F_ k(x)\), and \(G_ k(x)\) are known: \[ M_ 1(x)\sim Cx(\log x)^{-1}(\log\log x)^{D-1}, \] where \(D\) is Davenport’s constant depending only on the class group of \(R\), and \(C\) is a positive constant; \((\dots)\) \[ F_ k(x)\sim Cx(\log x)^{-1+(1/h)}(\log\log x)^{a_ k}, \] where \(h\) is the class number of \(R\) and \(a_ k\) is a combinatorial constant depending only on \(k\) and the class group of \(R\); \(C\) is a positive constant depending on \(R\) and \(k\). \((\dots)\) \[ G_ k(x)\sim Cx(\log x)^{-1+\mu}(\log\log x)^{d_ k}, \] where \(\mu\) and \(d_ k\) are combinatorial constants; \(\mu\) depends only on the class group of \(R\) and \(d_ k\) depends only on \(k\) and the class group of \(R\); \(C\) is a positive constant depending on \(R\) and \(k.(\dots)\)
The purpose of this paper is twofold: First, we show that quantitative results concerning non-unique factorization can be derived in an abstract setting, which is applicable not only to rings of integers in algebraic number fields, but also to holomorphy rings in algebraic function fields over finite fields and to Hilbert semigroups and their generalizations. In the second place, we focus our attention on holomorphy rings in algebraic function fields over finite fields in order to obtain more precise asymptotic results.
In \(\S 2\) we introduce the notion of a formation as the appropriate setting for an abstract theory of non-unique factorization, and we study some Dirichlet series which are connected with our counting problems. In \(\S 3\) we investigate the sets \(M_ k\), \(F_ k\) and \(G_ k\) for arbitrary formations; in particular, we prove that the above mentioned asymptotic results for algebraic number fields remain valid in the general context. \((\dots)\) In \(\S 4\) we collect the necessary arithmetical facts on holomorphy rings in algebraic function fields. \((\dots)\) In \(\S 5\) we derive the analytic machinery which is needed for the proof of the main results in \(\S 6\).”.

MSC:
11R27 Units and factorization
11N45 Asymptotic results on counting functions for algebraic and topological structures
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