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 semi-algebraic set in the max-plus sense. In particular, it is a finite union of polyhedral sets

The Set of Realizations of a Max-Plus Linear Sequence Is Semi-Polyhedral

We show that the set of realizations of dimension n of a max-plus linear sequence is a nite 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 coecients of a commutative rational expression in one letter that yield a given max-plus rational series, is a nite union of polyhedral sets