L'Algebre Tropicale Comme Algebre De la Caracteristique 1 : Algebre Lineaire Sur Les Semi-Corps Idempotents

TL;DR

This paper develops a formal framework for linear algebra over idempotent semi-fields like tropical algebras, introducing new concepts such as singular points, kernels, and tropical dimension, with applications across various mathematical fields.

Contribution

It re-examines linear algebra over idempotent semi-fields and introduces novel notions like singular points, kernels, and tropical dimension, enhancing the theoretical foundation.

Findings

Redefinition of singular points as a generalization of zero

Introduction of the kernel of a linear form in tropical algebra

Application of duality to define tropical dimension

Abstract

We define a formal framework for the study of algebras of type Max-plus, Min-Plus, tropical algebras, and more generally algebras over a commutative idempotent semi-field. This work is motivated by the increasingly diversified use of these algebras which occur also in control theory, automata theory as well as in algebraic geometry, and in more specific ways in other parts of mathematics such as the theory of monoids. In this first article, we expecially re-examine linear algebra over idempotent semi-fields: the most delicate, but undoubtedly the most interesting point is the notion of a singular point seen as a generalization of the notion of zero. We thus rediscover many notions of regularity already introduced for matrices, and this permits us to define further notions, new in this context, such as that of the kernel of a linear form, and to apply duality to obtain a good notion of tropical dimension of a submodule.