# On rational series in one variable over certain dioids

Abstract : We give a characterization of rational series in one variable over certain idempotent semirings (commutative dioids) such as for instance the $(\max,+)$ ''semiring. We show that a series is rational iff it is merge of ultimately geometric series. As a by-product, we obtain a new proof of the periodicity theorem for powers of irreducible matrices and also some more general auxiliary results. We apply this characterization of rational series to the minimal realization problem for which we obtain an upper bound. We also obtain a lower bound in terms of minors in a symmetrized semiring
Type de document :
Rapport
[Research Report] RR-2162, INRIA. 1994
Domaine :

https://hal.inria.fr/inria-00074510
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 15:34:23
Dernière modification le : mardi 17 avril 2018 - 11:28:42
Document(s) archivé(s) le : dimanche 4 avril 2010 - 22:18:48

### Identifiants

• HAL Id : inria-00074510, version 1

### Citation

Stéphane Gaubert. On rational series in one variable over certain dioids. [Research Report] RR-2162, INRIA. 1994. 〈inria-00074510〉

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