The set of realizations of a max-plus linear sequence is semi-polyhedral

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coefficients of a commutative rational expression in one letter that yield a given max-plus linear sequence is a finite union of polyhedral sets. Research Report RRLIP 2010-33 Email addresses: [email protected] (Vincent Blondel ), [email protected] (Stéphane Gaubert ), [email protected] (Natacha Portier ) This work was partly supported by a grant Tournesol (Programme de coopération scientifique entre la France et la communauté Française de Belgique), and by the European Community Framework IV program through the research network ALAPEDES (“The Algebraic Approach to Performance Evaluation of Discrete Event Systems”). This work was partially funded by European Community under contract PIOF-GA-2009236197 of the 7th PCRD. Preprint submitted to Elsevier October 19, 2010