Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Tropical noetherity and Gröbner bases


Authors: Ya. Kazarnovskiĭ and A. G. Khovanskiĭ
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 797-811
MSC (2010): Primary 16S34
DOI: https://doi.org/10.1090/spmj/1359
Published electronically: July 27, 2015
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A set that is a Gröbner basis for an ideal with respect to every Gröbner ordering is called a universal Gröbner basis for that ideal. In the paper, it is proved that there exists a universal Gröbner basis in which the polynomials have controlled degrees. The main result is the theorem on the tropical Noetherity of a ring of Laurent polynomials. This theorem is close to the existence theorem for a universal basis and is needed for the tropical intersection theory in  $ (\mathbb{C}^*)^n$, which will be presented in a forthcoming paper.


References [Enhancements On Off] (What's this?)

  • 1. C. De Concini and C. Procesi, Complete symmetric varieties. II, Intersection theory, Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 481-513. MR 803344 (87a:14038)
  • 2. C. De Concini, Equivariant embeddings of homogeneous spaces, Proc. Intern. Congress of Mathematicians, vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 369-377. MR 934236 (89e:14045)
  • 3. W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335-353. MR 1415592 (97h:14070)
  • 4. B. Ya. Kazarnovskiĭ, Truncations of systems of equations ideals and varietes, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 3, 119-132; English transl., Izv. Math. 63 (1999), no. 3, 535-547. MR 1712124 (2000j:13053)
  • 5. A. Seidenberg, On the length of a Hilbert ascending chain, Proc. Amer. Math. Soc. 29 (1971), 443-450. MR 0280473 (43:6193)
  • 6. -, Constructive proof of Hilbert's theorem on ascending chains, Trans. Amer. Math. Soc. 174 (1972), 305-312. MR 0314829 (47:3379)
  • 7. G. Moreno-Socias, Length of polynomial ascending chains and primitive recursiveness, Math. Scand. 71 (1992), no. 2, 181-205. MR 1212703 (94d:13019)
  • 8. B. Ya. Kazarnovskiĭ, $ c$-fans and Newton polyhedra of algebraic varietes, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 3, 23-44; English transl., Izv. Math. 67 (2003), no. 3, 439-460. MR 1992192 (2005a:14072)
  • 9. G. Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634. MR 1988122 (2004d:14077)
  • 10. 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 (2006i:14058)
  • 11. I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical algebraic geometry, 2nd ed., Birkhauser, Basel, 2009. MR 2508011 (2010d:14086)
  • 12. B. Ya. Kazarnovskiĭ and A. G. Khovanskiĭ, Algebra and tropical geometry, 2011-2013, pp. 1-42. (to appear).
  • 13. S. P. Chulkov and A. G. Khovanskiĭ, Geometry of the semigroup $ \mathbb{Z}^n_{\geq 0}$. Applications to combinatorics, algebra and differential equations, MCNMO, M., 2006. (Russian)
  • 14. F. Mora and L. Robbiano, The Gröbner fan of an ideal. Computational aspects of commutative algebra, J. Symbolic Comput. 6 (1988), no. 2-3, 183-208. MR 988412 (90d:13004)
  • 15. V. Weispfenning, Constructing universal Gröbner bases, Lecture Notes in Comput. Sci., vol. 356, Springer, Berlin, 1989, pp. 408-417. MR 1008554 (91e:13029)
  • 16. N. Schwartz, Stability of Gröbner bases, J. Pure Appl. Algebra 53 (1988), no. 1-2, 171-186. MR 955616 (89i:13024)
  • 17. B. Ya. Kazarnovskiĭ and A. G. Khovanskiĭ, The universal Gröbner bases, Proc. Intern. Conf. on Polynomial Computer Algebra, St. Peterburg, 2011, pp. 65-69. (Russian)
  • 18. F. S. Macaulay, The algebraic theory of modular systems, Cambridge Tracts in Math. and Math. Phys., vol. 19, Cambridge Univ. Press, Cambridge, 1916. MR 1281612 (95i:13001)
  • 19. Th. Sauer, Gröbner bases, $ H$-bases and interpolations, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2293-2308. (electronic) MR 1814071 (2002b:13035)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 16S34

Retrieve articles in all journals with MSC (2010): 16S34


Additional Information

Ya. Kazarnovskiĭ
Affiliation: Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia
Email: kazbori@gmail.com

A. G. Khovanskiĭ
Affiliation: Institute for Systems Analysis, Russian Academy of Sciences, 60-letiya Oktyabrya pr. 9, Moscow 117312; Independent University of Moscow, Bolshoy Vlasyevskiǐ Pereulok 11, Moscow 119002, Russia; University Of Toronto, Canada
Email: askold@math.toronto.edu

DOI: https://doi.org/10.1090/spmj/1359
Keywords: Laurent polynomial, ideal, tropical basis, universal Gr\"obner basis, Seidenberg theorem
Received by editor(s): October 17, 2013
Published electronically: July 27, 2015
Additional Notes: The first author was partially supported by the grant SS-4850.2012.1; the second author was partially supported by the Canadian grant 0GP0156833
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society