A tropical nullstellensatz

Shustin, Eugenii; Izhakian, Zur

doi:10.1090/S0002-9939-07-09005-3

A tropical nullstellensatz
HTML articles powered by AMS MathViewer

by Eugenii Shustin and Zur Izhakian PDF

Proc. Amer. Math. Soc. 135 (2007), 3815-3821 Request permission

Abstract:

We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and summation.

References

Amoebas and tropical geometry. Preprint available at http://www.aimath.org/WWN/ amoebas.
M. Einsiedler, M. Kapranov, and D. Lind, Non-Archimedean amoebas and tropical varieties. Preprint at arXiv:math.AG/0408311.
I. Itenberg, Amibes des variétés algébriques et denombrement de courbes [d’après G. Mikhalkin]. Séminaire N. Bourbaki 921, vol. 2002-03, Juin 2003.
Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, Welschinger invariant and enumeration of real rational curves, Int. Math. Res. Not. 49 (2003), 2639–2653. MR 2012521, DOI 10.1155/S1073792803131352
I. V. Itenberg, V. M. Kharlamov, and E. I. Shustin, Logarithmic equivalence of the Welschinger and the Gromov-Witten invariants, Uspekhi Mat. Nauk 59 (2004), no. 6(360), 85–110 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 6, 1093–1116. MR 2138469, DOI 10.1070/RM2004v059n06ABEH000797
Z. Izhakian, Tropical Arithmetic and Algebra of Tropical Matrices. Preprint at arXiv:math.AG/0505458.
V. N. Kolokoltsov and V. P. Maslov, Idempotent Analysis and Applications. Kluwer Acad. Publ., Dordrecht, The Netherlands, 1997.
Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 1882331, DOI 10.1142/9789812799821_{0}007
M. Kontsevich, M. and Yu. Tschinkel, Nonarchimedean Kähler geometry. Preprint, 2002.
Grigory Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634 (English, with English and French summaries). MR 1988122, DOI 10.1016/S1631-073X(03)00104-3
Grigory Mikhalkin, Amoebas of algebraic varieties and tropical geometry, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 257–300. MR 2102998, DOI 10.1007/0-306-48658-X_{6}
Grigory Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), no. 5, 1035–1065. MR 2079993, DOI 10.1016/j.top.2003.11.006
Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI 10.1090/S0894-0347-05-00477-7
E. Shustin, A tropical approach to enumerative geometry, Algebra i Analiz 17 (2005), no. 2, 170–214; English transl., St. Petersburg Math. J. 17 (2006), no. 2, 343–375. MR 2159589, DOI 10.1090/S1061-0022-06-00908-3
David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, DOI 10.1515/advg.2004.023
D. Speyer and B. Sturmfels, Tropical Mathematics. Preprint arXiv:math.CO/0408099.
Oleg Viro, Dequantization of real algebraic geometry on logarithmic paper, European Congress of Mathematics, Vol. I (Barcelona, 2000) Progr. Math., vol. 201, Birkhäuser, Basel, 2001, pp. 135–146. MR 1905317