On Rational Series in One Variable Over Certain Dioids

The document characterized rational series in one variable over certain idempotent semirings, such as the "(max,+)" semiring, by showing that a series was rational if and only if it was a merge of ultimately geometric series. This characterization led to a new proof of the periodicity theorem for powers of irreducible matrices and provided additional auxiliary results. The study also applied this characterization to the minimal realization problem, establishing both an upper bound and a lower bound in terms of minors in a symmetrized semiring.