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Bounds on the Number of Connected Components for Tropical Prevarieties

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Abstract

For a tropical prevariety in \({\mathbb {R}}^n\) given by a system of k tropical polynomials in n variables with degrees at most d, we prove that its number of the connected components is less than \({k+7n-1 \atopwithdelims ()3n} \cdot \frac{d^{3n}}{k+n+1}\). On a number of 0-dimensional connected components a better bound \({k \atopwithdelims ()n} \cdot \frac{d^n}{k-n+1}\) is obtained, which extends the Bezout bound due to B. Sturmfels from the case \(k=n\) to an arbitrary \(k\ge n\). Also we show that the latter bound is close to sharp, in particular, the number of connected components can depend on k.

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Acknowledgements

The second author is grateful to the Grant RSF 16-11-10075 and to MCCME for wonderful working conditions and inspiring atmosphere. Both authors are grateful to the Max-Planck Institut für Mathematik, Bonn for its hospitality during writing this paper and to anonymous referees whose remarks helped to improve the exposition.

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  1. Academic University, Ulitsa Khlopina 8, Saint Petersburg, Russia, 194021

    Alex Davydow

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

    Dima Grigoriev

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  1. Alex Davydow
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  2. Dima Grigoriev
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Correspondence to Dima Grigoriev.

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Davydow, A., Grigoriev, D. Bounds on the Number of Connected Components for Tropical Prevarieties. Discrete Comput Geom 57, 470–493 (2017). https://doi.org/10.1007/s00454-016-9839-6

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  • DOI: https://doi.org/10.1007/s00454-016-9839-6

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