Dominique Castella | Searcharxiv

arXiv2008

L'Algebre Tropicale Comme Algebre De la Caracteristique 1 : Polynomes Rationnels Et Fonctions Polynomiales

We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of tropical geometry which appears to be the algebraic geometry of tropical algebra. In fact, the most interesting object is the image of a polynomial algebra in its semi-field of fractions. We can thus obtain, over good semi-fields, the analog of classical correpondences between polynomials, and varieties of zeros... For example, we show that the algebras of polynomial functions over a tropical curves associated to a polynomial P, is, as in classical algebraic geometry, the quotient of the polynomial algebra by the ideal generated by P.

math.RA↗

arXiv2008

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

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.

math.AC↗