The Frobenius problem: a geometric approach
For the well known Frobenius problem, we present a new geometric approach, based on the use of the $n$-dimensional lattice $\mathbb{Z}^n$, where $n$ is the number of generators. Within this approach we are able to study the cases of two and three generators. The main feature of our geometric represe...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2007 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/1170 |
| Acceso en línea: | https://hdl.handle.net/2117/1170 |
| Access Level: | acceso abierto |
| Palabra clave: | Representations of semigroups Diophantine equations Frobenius problem integer lattice semigroup Semigrups Equacions diofàntiques Classificació AMS::11 Number theory::11D Diophantine equations Classificació AMS::20 Group theory and generalizations::20M Semigroups |
| Sumario: | For the well known Frobenius problem, we present a new geometric approach, based on the use of the $n$-dimensional lattice $\mathbb{Z}^n$, where $n$ is the number of generators. Within this approach we are able to study the cases of two and three generators. The main feature of our geometric representation is that we can nicely visualize the set of {\em gaps}, i.e., the non-representable positive integers. In the case of two generators, we give a description of the set of gaps. Moreover, for any positive integer, $m$, we derive a simple expression for the denumerant $d(m;a,b)$. We show that we can use the $2$-dimensional lattice associated to the set of generators $\{ a,b\}$ to study the Frobenius problem with generators $\{ a,b,c\}$. In particular, we give, as for two generators, a graphical representation of the set of gaps. For a large set of possible values of $c$, this representation allows us to simplify the computation of the Frobenius number and compute the number of gaps. |
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