A construction of integer-valued polynomials with prescribed sets of lengths of factorizations |
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Authors: | Sophie Frisch |
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Institution: | 1. Institut für Mathematik A, Technische Universit?t Graz, Steyrergasse 30, 8010, Graz, Austria
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Abstract: | For an arbitrary finite non-empty set $S$ of natural numbers greater $1$ , we construct $f\in \text{ Int }(\mathbb{Z })=\{g\in \mathbb{Q }x]\mid g(\mathbb{Z })\subseteq \mathbb{Z }\}$ such that $S$ is the set of lengths of $f$ , i.e., the set of all $n$ such that $f$ has a factorization as a product of $n$ irreducibles in $\text{ Int }(\mathbb{Z })$ . More generally, we can realize any finite non-empty multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of $f$ . |
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