Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Bihan, F.: Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems. Discret. Comput. Geom. 55(4), 907–933 (2016)
Bihan, F., Sottile, F.: Betti number bounds for fewnomial hypersurfaces via stratified Morse theory. Proc. Am. Math. Soc. 137(5), 2825–2833 (2009)
Björner, A., Las Verginas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Cambridge University Press, Cambridge (1999)
Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007)
Davydow, A., Grigoriev, D.: Bounds on the number of connected components for tropical prevarieties. Discret. Comput. Geom. 57(2), 470–493 (2017)
Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)
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)
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)
Grigoriev, D., Podolskii, V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex. 24(1), 31–64 (2015)
Hardt, R.M.: Semi-algebraic local triviality in semi-algebraic mappings. Am. J. Math. 102, 291–302 (1980)
Khovanskii, A.: Fewnomials, Translations of Mathematical Monographs, vol. 88. American Mathematical Society, Providence (1991)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society, Providence (2015)
Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1964)
Prasolov, V.V.: Elements of Combinatorial and Differential Topology. American Mathematical Society, Providence (2006)
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)
Steffens, R., Theobald, T.: Combinatorics and genus of tropical intersections and Ehrhart theory. SIAM J. Discret. Math. 24, 17–32 (2010)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grigoriev, D., Vorobjov, N. Upper Bounds on Betti Numbers of Tropical Prevarieties. Arnold Math J. 4, 127–136 (2018). https://doi.org/10.1007/s40598-018-0086-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-018-0086-1