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

TL;DR

This paper explores the structure of polynomial algebras over idempotent semi-fields within tropical algebra, establishing analogies with classical algebraic geometry and analyzing the algebraic properties of tropical polynomial functions.

Contribution

It extends the understanding of polynomial algebras in tropical geometry, especially over semi-fields, and demonstrates their correspondence with classical algebraic structures.

Findings

Polynomial algebras over tropical semi-fields mirror classical quotient structures.

The image of polynomial algebras in semi-fields of fractions relates to tropical varieties.

Tropical polynomial functions form algebras analogous to classical polynomial function algebras.

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.