The Frobenius problem over real number fields

Abstract

Given a number field $K$ that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers ${\mathfrak{O}}_K$ of $K$ by describing certain Frobenius semigroups, $\mathrm{Frob}({\alpha }_1,\dots ,{\alpha }_n)$, for appropriate elements ${\alpha }_1,\dots ,{\alpha }_n\in {\mathfrak{O}}_K$. We construct a partial ordering on $\mathrm{Frob}({\alpha }_1,\dots ,{\alpha }_n)$, and show that this set is completely described by the maximal elements with respect to this ordering. We also show that $\mathrm{Frob}({\alpha }_1,\dots ,{\alpha }_n)$ will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as $n$ is fixed and ${\alpha }_1,\dots ,{\alpha }_n\in {\mathfrak{O}}_K$ vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.

17 pages