Tropical Effective Primary and Dual Nullstellens"atze

Authors Dima Grigoriev, Vladimir V. Podolskii

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Dima Grigoriev
Vladimir V. Podolskii

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Dima Grigoriev and Vladimir V. Podolskii. Tropical Effective Primary and Dual Nullstellens"atze. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 379-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove tropical Nullstellensatz and moreover we show effective formulation of this theorem. Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.
  • tropical algebra
  • tropical geometry
  • Nullstellensatz


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  1. Marianne Akian, Stephane Gaubert, and Alexander Guterman. Linear independence over tropical semirings and beyond. Contemporary Mathematics, 495:1-33, 2009. Google Scholar
  2. Marianne Akian, Stephane Gaubert, and Alexander Guterman. Tropical polyhedra are equivalent to mean payoff games. International Journal of Algebra and Computation, 22(1), 2012. Google Scholar
  3. Peter Butkovič. Max-linear Systems: Theory and Algorithms. Springer, 2010. Google Scholar
  4. M. Develin, F. Santos, and B. Sturmfels. On the rank of a tropical matrix. Combinatorial and computational geometry, 52:213-242, 2005. Google Scholar
  5. Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind. Non-archimedean amoebas and tropical varieties. Journal fur die reine und angewandte Mathematik (Crelles Journal), 2006.601:139-157, 2007. Google Scholar
  6. Marc Giusti, Joos Heintz, and Juan Sabia. On the efficiency of effective Nullstellensätze. Computational complexity, 3(1):56-95, 1993. Google Scholar
  7. Dima Grigoriev. On a tropical dual Nullstellensatz. Advances in Applied Mathematics, 48(2):457 - 464, 2012. Google Scholar
  8. Dima Grigoriev. Complexity of solving tropical linear systems. Computational Complexity, 22(1):71-88, 2013. Google Scholar
  9. Dima Grigoriev and Vladimir V. Podolskii. Tropical effective primary and dual Nullstellensätze. CoRR, abs/1409.6215, 2014. Google Scholar
  10. Dima Grigoriev and Vladimir V. Podolskii. Complexity of tropical and min-plus linear prevarieties. Computational complexity, to appear, pages 1-34, 2015. Google Scholar
  11. Dima Grigoriev and Vladimir Shpilrain. Tropical cryptography. Communications in Algebra, 42(6):2624-2632, 2014. Google Scholar
  12. Birkett Huber and Bernd Sturmfels. A polyhedral method for solving sparse polynomial systems. Mathematics of Computation, 64:1541-1555, 1995. Google Scholar
  13. I. Itenberg, G. Mikhalkin, and E.I. Shustin. Tropical Algebraic Geometry. Oberwolfach Seminars. Birkhäuser, 2009. Google Scholar
  14. Zur Izhakian. Tropical algebraic sets, ideals and an algebraic Nullstellensatz. International Journal of Algebra and Computation, 18(06):1067-1098, 2008. Google Scholar
  15. Zur Izhakian and Louis Rowen. The tropical rank of a tropical matrix. Communications in Algebra, 37(11):3912-3927, 2009. Google Scholar
  16. Stasys Jukna. Lower bounds for tropical circuits and dynamic programs. Electronic Colloquium on Computational Complexity (ECCC), 21:80, 2014. Google Scholar
  17. János Kollár. Sharp effective Nullstellensatz. J. Amer. Math. Soc., 1:963-975, 1988. Google Scholar
  18. D. Lazard. Algèbre linéaire sur K[X₁,…,X_n] et élimination. Bull. Soc. Math. France, 105(2):165-190, 1977. Google Scholar
  19. D. Lazard. Resolution des systemes d'equations algebriques. Theoret. Comput. Sci., 15(1):77-110, 1981. Google Scholar
  20. Diane Maclagan and Bernd Sturmfels. Introduction to Tropical Geometry, volume 161 of AMS Graduate Studies in Mathematics. AMS, to appear, 2015. Google Scholar
  21. Grigory Mikhalkin. Amoebas of algebraic varieties and tropical geometry. In Simon Donaldson, Yakov Eliashberg, and Mikhael Gromov, editors, Different Faces of Geometry, volume 3 of International Mathematical Series, pages 257-300. Springer US, 2004. Google Scholar
  22. Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, 377:289-317, 2003. Google Scholar
  23. Eugenii Shustin and Zur Izhakian. A tropical Nullstellensatz. Proceedings of the American Mathematical Society, 135(12):3815-3821, 2007. Google Scholar
  24. Bernd Sturmfels. Solving Systems of Polynomial Equations, volume 97 of CBMS Regional Conference in Math. American Mathematical Society, 2002. Google Scholar
  25. Luis Felipe Tabera. Tropical resultants for curves and stable intersection. Revista Matemática Iberoamericana, 24(3):941-961, 04 2008. Google Scholar
  26. Thorsten Theobald. On the frontiers of polynomial computations in tropical geometry. J. Symb. Comput., 41(12):1360-1375, 2006. Google Scholar