Arnold Mathematical Journal

, Volume 4, Issue 1, pp 127–136 | Cite as

Upper Bounds on Betti Numbers of Tropical Prevarieties

  • Dima Grigoriev
  • Nicolai Vorobjov
Research Contribution


We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials.


Tropical prevariety Betti numbers Polyhedral complex 

Mathematics Subject Classification




Authors thank D. Feichtner-Kozlov, G.M. Ziegler, and the anonymous referee for useful comments. Part of this research was carried out during authors’ joint visit in September 2017 to the Hausdorff Research Institute for Mathematics at Bonn University, under the program Applied and Computational Algebraic Topology, to which they are very grateful. D. Grigoriev was partly supported by the RSF Grant 16-11-10075.


  1. Bertrand, B., Bihan, F.: Intersection multiplicity numbers between tropical hypersurfaces. In: Algebraic and Combinatorial Aspects of Tropical Geometry. Contemporary Mathematics, vol. 589, pp. 1–19. American Mathematical Society, Providence (2013)Google Scholar
  2. Bihan, F.: Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems. Discret. Comput. Geom. 55(4), 907–933 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bihan, F., Sottile, F.: Betti number bounds for fewnomial hypersurfaces via stratified Morse theory. Proc. Am. Math. Soc. 137(5), 2825–2833 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Björner, A., Las Verginas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  5. Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Davydow, A., Grigoriev, D.: Bounds on the number of connected components for tropical prevarieties. Discret. Comput. Geom. 57(2), 470–493 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gabrielov, A., Vorobjov, N.: Approximation of definable sets by compact families, and upper bounds on homotopy and homology. J. Lond. Math. Soc. 2(80), 35–54 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gabrielov, A., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian functions. In: Ilyashenko, Yu., Rousseau, C. (eds.) Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, pp. 211–250. Kluwer, Dordrecht (2004)CrossRefGoogle Scholar
  10. Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex. 24(1), 31–64 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hardt, R.M.: Semi-algebraic local triviality in semi-algebraic mappings. Am. J. Math. 102, 291–302 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Khovanskii, A.: Fewnomials, Translations of Mathematical Monographs, vol. 88. American Mathematical Society, Providence (1991)Google Scholar
  13. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society, Providence (2015)CrossRefzbMATHGoogle Scholar
  14. Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Prasolov, V.V.: Elements of Combinatorial and Differential Topology. American Mathematical Society, Providence (2006)CrossRefzbMATHGoogle Scholar
  16. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Litvinov, G., Maslov, V. (eds.) Idempotent Mathematics and Mathematical Physics (Proceedings Vienna 2003). Contemporary Mathematics, vol. 377, pp. 289–317. American Mathematical Society, Providence (2005)CrossRefGoogle Scholar
  17. Steffens, R., Theobald, T.: Combinatorics and genus of tropical intersections and Ehrhart theory. SIAM J. Discret. Math. 24, 17–32 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Zaslavski, T.: Facing up to arrangements: face-count formulas for partition of space by hyperplanes. vol. 1, issue 154, 102pp. Memoirs of the American Mathematical Society (1975)Google Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2018

Authors and Affiliations

  1. 1.CNRS, MathématiquesUniversité de LilleVilleneuve d’AscqFrance
  2. 2.Department of Computer ScienceUniversity of BathBathUK

Personalised recommendations