Polynomials over idempotent semifields

Abstract

We study univariate polynomials with coefficients in an idempotent semifield and their factorization. We do not assume the idempotent semifield under consideration to be totally ordered, in contrast with most of the existing work on this topic. We notably determine when a polynomial splits into linear factors, and when its associated polynomial function does so. These results lead us to characterize algebraically closed idempotent semifields -- those in which every polynomial function splits. We prove in particular that every complete idempotent semifield is algebraically closed. We also relate algebraic closedness to the properties of preradicability and radicability and to the existence of solutions to polynomial equations or inequalities.