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

Abstract

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.