Algèbres de polynômes tropicaux
[Tropical polynomial algebras]
Dominique Castella
Annales Mathématiques Blaise Pascal, Volume 20 (2013) no. 2, p. 301-330

We continue, in this second article, the study of 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 correspondences between polynomials, polynomial functions and varieties of zeros...For example, we show that the algebras of polynomial functions over a tropical curve associated to a polynomial P, is, as in classical algebraic geometry, the quotient of the polynomial algebra by the radical of the ideal generated by P and we give a purely algebraic complete description of this ideal (i. e. a new demonstration of the Tropical Nullstellensatz obtained previously by Izhakian, Shustin et Rowen).

Nous continuons dans ce second article, l’étude des outils algébrique de l’algèbre de la caractéristique 1 : nous examinons plus spécialement ici les algèbres de polynômes sur un semi-corps idempotent. Ce travail est motivé par le développement de la géométrie tropicale qui apparaît comme étant la géométrie algébrique de l’algèbre tropicale. En fait l’objet algébrique le plus intéressant est l’image de l’algèbre de polynôme dans son semi-corps des fractions. Nous pouvons ainsi retrouver sur les bons semi-corps l’analogue des correspondances classiques entre polynômes, fonctions polynomiales et ensemble de zéros...Par exemple, nous montrons que l’algèbre des fonctions polynomiales sur une hypersurface tropicale associée à un polynôme P, est comme dans le cas classique, le quotient de l’algèbre de polynômes par le radical de l’idéal engendré par P et nous retrouvons ainsi, de façon purement algébrique la description complète de cet idéal (i.e. une nouvelle démonstration du Tropical Nullstellensatz obtenu par Izhakian, Shustin et Rowen). Ces méthodes devraient permettre d’obtenir des algorithmes de factorisation pour les polynômes tropicaux.

DOI : https://doi.org/10.5802/ambp.328
Classification:  55B20,  14A05,  16Y60
Keywords: Polynomial algebra, tropical algebra, idempotent semi-fields, tropical geometry
@article{AMBP_2013__20_2_301_0,
     author = {Castella, Dominique},
     title = {Alg\`ebres de polyn\^omes tropicaux},
     journal = {Annales Math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {20},
     number = {2},
     year = {2013},
     pages = {301-330},
     doi = {10.5802/ambp.328},
     zbl = {1311.14060},
     mrnumber = {3138031},
     zbl = {06251802},
     language = {fr},
     url = {https://ambp.centre-mersenne.org/item/AMBP_2013__20_2_301_0}
}
Castella, Dominique. Algèbres de polynômes tropicaux. Annales Mathématiques Blaise Pascal, Volume 20 (2013) no. 2, pp. 301-330. doi : 10.5802/ambp.328. https://ambp.centre-mersenne.org/item/AMBP_2013__20_2_301_0/

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