Algèbres de polynômes tropicaux

Castella, Dominique

Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 301-330.

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.

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).

[1] Akian, M.; Bapat, R.; Gaubert, S. Max-plus algebras, Chapter 25 in the Handbook of Linear Algebra,, Discrete Mathematics and Its Applications, Volume 39,, Chapman and Hall, 2007

[2] Aroca, F. Krull-tropical hypersurfaces, Annales de la faculté des sciences de Toulouse (2010), pp. 525-538 | DOI | Numdam | MR | Zbl

[3] Banerjee, S. Tropical geometry over higher dimensional local fields, arXiv : 1105.5873 v2 (2012)

[4] Castella, D. L’algèbre tropicale comme algèbre de la caractéristique 1 : Polynômes rationnels et fonctions polynomiales, arXiv : 0809.0231 (2008)

[5] Castella, D. Eléments d’algèbre linéaire tropicale, Linear Algebra and Its Applications, Volume 432 (2010), pp. 1460-1474 | DOI | MR | Zbl

[6] Grigoriev, D. On a tropical dual Nullstellensatz, Adv. Appl. Math., Volume 48 (2012), pp. 457-464 | DOI | MR | Zbl

[7] Hebisch, U.; Weinert, H.J. Semirings. Algebraic theory and application in computer sciences, World scientific, Singapore, 1998 | MR | Zbl

[8] Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii Tropical algebraic geometry, Oberwolfach Seminars, 35, Birkhäuser Verlag, Basel, 2009 | DOI | MR | Zbl

[9] Izhakian, Z. Tropical algebraic sets ideals and an algebraic Nullstellensatz, IJAC, Volume 18(6) (2008), pp. 1067-1098 | MR | Zbl

[10] Izhakian, Z.; Rowen, L. The tropical rank of a tropical matrix, Comm. in Algebra, Volume 37 (2009), pp. 3912-3927 | DOI | MR | Zbl

[11] Izhakian, Z.; Rowen, L. Supertropical algebra, Adv. in Math., Volume 225 (2010), pp. 2222-2286 | DOI | MR

[12] Kolokoltsov, V. N.; Maslov, V. P. Idempotent analysis and its applications, Kluver academic publishers, Dordrecht, 1997 | MR | Zbl

[13] Lescot, P. Absolute algebra II Ideals and spectra, J. of Pure and Applied Algebra, Volume 215(7) (2011), pp. 1782-1790 | DOI | MR | Zbl

[14] Litvinov, G. L.; Maslov, V. P. Idempotent Mathematics and Mathematical Physics, Number 377, Contemp. Math. Amer. Math. Soc., 2005 | MR | Zbl

[15] Mikhalkin, G.; Int. Math. Ser.(N.Y.), vol.3 Amoebas of algebraic varieties and tropical geometry, Different faces of geometry, Kluwer/Plenum, NewYork, 2004, pp. 257-300 | MR | Zbl

[16] Mikhalkin, G. Decomposition into pairs-of-plants for complex algebraic hypersurfaces, Topology, Volume 43(5) (2004), pp. 1035-1065 | DOI | MR | Zbl

[17] Richter-Gebert, J.; Sturmfels, B.; Theobland, T. First steps in tropical geometry, Contemporary Mathematics, Volume 377 (2005), pp. 289-317 | DOI | MR | Zbl

[18] Shustin, E.; Izhakian, Z. A tropical Nullstellensatz, Proc. Amer. Math. Soc., Volume 135 (12) (2007), pp. 3815-3821 | DOI | MR | Zbl

[19] Speyer, D.E.; Sturmfels, B. Tropical mathematics, Mathematics Magazine, Volume 82 (3) (2009), pp. 163-173 | DOI | MR | Zbl

[20] Tabera, L. Tropical resultants for curves and stable intersection, Rev. Mat. Iberoam, Volume 24 (2008), pp. 941-961 | DOI | MR | Zbl

[21] Viro, O.; 201, Progress Math. Dequantization of real algebraic geometry on logarithm, European Congress of Mathematics, Vol. I, Barcelone, Birkhäuser, 2001, pp. 135-146