Publikation: Characterizing absolutely irreducible i...
Stammdaten
| Titel: | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| Untertitel: | |
| Kurzfassung: | Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials $F \in \Int(R)$ where $R$ is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number $S\in \N$ that reduces the absolute irreducibility of $F$ to the unique factorization of $F^S$. To this end, we establish a connection between the factors of powers of $F$ and the kernel of a certain linear map that we associate to $F$. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element $v$ of this kernel, we explicitly construct non-trivial factorizations of $F^k$, provided that $k\ge L$, where $L$ depends on $F$ as well as the choice of $v$. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for $k$, one of which only depends on the valuation of the denominator of $F$ and the size of the residue class field of $R$. |
| Schlagworte: | Non-unique factorization, irreducible elements, absolutely irreducible elements, integer-valued polynomials |
| Publikationstyp: | Beitrag in Zeitschrift (Autorenschaft) |
| Erscheinungsdatum: | 25.07.2023 (Online) |
| Erschienen in: |
Journal of Algebra
Journal of Algebra
(
Academic Press / Elsevier;
)
zur Publikation |
| Titel der Serie: | - |
| Bandnummer: | 533 |
| Heftnummer: | - |
| Erstveröffentlichung: | Ja |
| Version: | - |
| Seite: | S. 696 - 721 |
Versionen
| Keine Version vorhanden |
| Erscheinungsdatum: | 25.07.2023 |
| ISBN (e-book): | - |
| eISSN: | - |
| DOI: | http://dx.doi.org/10.1016/j.jalgebra.2023.06.026 |
| Homepage: | https://www.sciencedirect.com/science/article/pii/S0021869323003307?via%3Dihub |
| Open Access |
|
Zuordnung
| Organisation | Adresse | ||||
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Fakultät für Technische Wissenschaften
Institut für Mathematik
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AT - 9020 Klagenfurt am Wörthersee |
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| Forschungscluster | Kein Forschungscluster ausgewählt |
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| Organisation | Adresse | ||
|---|---|---|---|
Kabale University, Department of Mathematics
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UG Kabale |
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