Upper Bounds on Betti Numbers of Tropical Prevarieties
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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.
KeywordsTropical prevariety Betti numbers Polyhedral complex
Mathematics Subject Classification14T05
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.
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