Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements. (English) Zbl 07895595
For an integral domain \(R\) with fraction field \(K\) denote by Int\((R)\) the ring of integer-valued polynomials of \(R\), i.e. the ring of polynomials over \(K\) with \(f(R)\subset R\). It has been shown by S. Frisch [Monatsh. Math. 171, No. 3–4, 341–350 (2013; Zbl 1282.13004)] that if \(k\) and \(n_1,n_2,\dots,n_k\) are given integers \(\ge2\), then there is a polynomial \(f(X)\in Int(Z)\) having \(k\) distinct factorizations into irreducibles of lengths \(n_1,n_2,\dots,n_k\). Later S. Frisch et al. [J. Algebra 528, 231–249 (2019; Zbl 1419.13002)] established the same assertion for Int\((R)\) in the case when \(R\) is a Dedekind domain having infinitely many prime ideals with finite residue fields, and V. Fadinger-Held et al. [Monatsh. Math. 202, No. 4, 773–789 (2023; Zbl 07753444)] did this when \(R\) is a valuation ring of a global field.
In the paper under review, the authors show that this assertion holds when \(R\) is a Krull domain having a prime ideal with finite residue field.
In the paper under review, the authors show that this assertion holds when \(R\) is a Krull domain having a prime ideal with finite residue field.
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13A05 | Divisibility and factorizations in commutative rings |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13P05 | Polynomials, factorization in commutative rings |
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