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Tropical Effective Primary and Dual Nullstellensätze

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Abstract

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 studying properties of mathematical objects such as algebraic varieties 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 a tropical Nullstellensatz, and moreover, we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.

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Notes

  1. To avoid a confusion we note that we use the word “dual” in two different meanings. First, we use it in the term “dual Nullstellensatz” as opposed to the standard version of Nullstellensatz. This means that the dual Nullstellensatz is obtained from the standard Nullstellensatz by the (linear) duality (see [11] and later on in this paper). Second, we use the word “dual” in the term “duality result” to denote the general type of results. Since the standard Nullstellensatz is a duality result itself, applying the linear duality to it results in a non-duality result. Thus, the dual Nullstellensatz is not a duality result in a proper sense, but rather the word “dual” is used in contrast to the customary Nullstellensatz which we name “primary”.

References

  1. Akian, M., Gaubert, S., Guterman, A.: Linear independence over tropical semirings and beyond. Contemp. Math. 495, 1–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. J. Algebra Comput. 22(1), 1250001 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertram, A., Easton, R.: The tropical nullstellensatz for congruences. Adv. Math. 308, 36–82 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogart, T., Jensen, A., Speyer, D., Sturmfels, B., Thomas, R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007). Effective Methods in Algebraic Geometry (MEGA 2005)

  5. Brownawell, W.D.: Bounds for the degrees in the Nullstellensatz. Ann. Math. 126(3), 577–591 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Butkovič, P.: Max-Linear Systems: Theory and Algorithms. Springer, London (2010)

    Book  MATH  Google Scholar 

  7. Davydow, A., Grigoriev, D.: Bounds on the number of connected components for tropical prevarieties. Discrete Comput. Geom. 57(2), 470–493 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. Combin. Comput. Geom. 52, 213–242 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Einsiedler, M., Kapranov, M., Lind, D.: Non-archimedean amoebas and tropical varieties. J. fur die reine und angew. Math. (Crelles J.) 2006(601), 139–157 (2007)

    MATH  Google Scholar 

  10. Giusti, M., Heintz, J., Sabia, J.: On the efficiency of effective Nullstellensätze. Comput. Complex. 3(1), 56–95 (1993)

    Article  MATH  Google Scholar 

  11. Grigoriev, D.: On a tropical dual Nullstellensatz. Adv. Appl. Math. 48(2), 457–464 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grigoriev, D.: Complexity of solving tropical linear systems. Comput. Complex. 22(1), 71–88 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex. 24(1), 31–64 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grigoriev, D., Podolskii, V.V.: Tropical effective primary and dual Nullstellensätze. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 379–391, Dagstuhl, Germany (2015). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik

  15. Grigoriev, D., Shpilrain, V.: Tropical cryptography. Commun. Algebra 42(6), 2624–2632 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grigoriev, D., Vorobjov, N.: Complexity of null-and positivstellensatz proofs. Ann. Pure Appl. Logic 113(1–3), 153–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64, 1541–1555 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry. In: Oberwolfach Seminars, Birkhäuser (2009)

  19. Izhakian, Z.: Tropical algebraic sets, ideals and an algebraic Nullstellensatz. Int. J. Algebra Comput. 18(06), 1067–1098 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Izhakian, Z., Rowen, L.: The tropical rank of a tropical matrix. Commun. Algebra 37(11), 3912–3927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Joó, D., Mincheva, K.: Prime congruences of additively idempotent semirings and a nullstellensatz for tropical polynomials. Sel. Math. (2017). https://doi.org/10.1007/s00029-017-0322-x

  22. Jukna, S.: Lower bounds for tropical circuits and dynamic programs. Electronic Colloquium on Computational Complexity (ECCC), 21:80 (2014)

  23. Kollár, J.: Sharp effective Nullstellensatz. J. Am. Math. Soc. 1, 963–975 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lazard, D.: Algèbre linéaire sur \({K}[{X}_1,\ldots,{X}_n]\) et élimination. Bull. Soc. Math. France 105(2), 165–190 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lazard, D.: Resolution des systemes d’equations algebriques. Theor. Comput. Sci. 15(1), 77–110 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  27. Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Donaldson, S., Eliashberg, Y., Gromov, M. (eds.) Different Faces of Geometry. International Mathematical Series, vol. 3, pp. 257–300. Springer, New York (2004)

    Chapter  Google Scholar 

  28. Purbhoo, K.: A nullstellensatz for amoebas. Duke Math. J. 141(3), 407–445 (2008)

  29. Ren, Q., Shaw, K., Sturmfels, B.: Tropicalization of del pezzo surfaces. Adv. Math. 300, 156–189 (2016). Special volume honoring Andrei Zelevinsky

  30. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. Idempotent Math. Math. Phys. Contemp. Math. 377, 289–317 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shustin, E., Izhakian, Z.: A tropical Nullstellensatz. Proc. Am. Math. Soc. 135(12), 3815–3821 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Steffens, R., Theobald, T.: Combinatorics and genus of tropical intersections and ehrhart theory. SIAM J. Discrete Math. 24(1), 17–32 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS Regional Conference in Mathematics, vol. 97. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  34. Tabera, L..F.: Tropical resultants for curves and stable intersection. Rev. Mate. Iberoam. 24(3), 941–961 (2008)

  35. Theobald, T.: On the frontiers of polynomial computations in tropical geometry. J. Symb. Comput. 41(12), 1360–1375 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vorobyev, N.: Extremal algebra of positive matrices. Elektron. Informationsverarbeitung und Kybern. 3, 39–71 (1967)

    MathSciNet  Google Scholar 

  37. Weispfenning, V.: The complexity of linear problems in fields. J. Symb. Comput. 5(1), 3–27 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author is grateful to the Grant RSF 16-11-10075 and to both MCCME and the Max-Planck Institut für Mathematik, Bonn for wonderful working conditions and an inspiring atmosphere.

The work of the second author is partially supported by the grant of the President of Russian Federation (MK-5379.2018.1) and by the Russian Academic Excellence Project ‘5-100’. Part of the work of the second author was done during the visit to Max-Planck Institut für Mathematik, Bonn.

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Authors and Affiliations

  1. CNRS, Mathématiques, Université de Lille, 59655 , Villeneuve-d’Ascq, France

    Dima Grigoriev

  2. Steklov Mathematical Institute, 119991, Moscow, Russia

    Vladimir V. Podolskii

  3. National Research University Higher School of Economics, 125319, Moscow, Russia

    Vladimir V. Podolskii

Authors
  1. Dima Grigoriev
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  2. Vladimir V. Podolskii
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Correspondence to Vladimir V. Podolskii.

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Editor in Charge János Pach

An extended abstract of a preliminary version [14] appeared in the proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015).

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Grigoriev, D., Podolskii, V.V. Tropical Effective Primary and Dual Nullstellensätze. Discrete Comput Geom 59, 507–552 (2018). https://doi.org/10.1007/s00454-018-9966-3

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  • DOI: https://doi.org/10.1007/s00454-018-9966-3

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