Real tropical hypersurfaces are directly related to T-hypersurfaces (piecewise-linear hypersurfaces arising in the combinatorial patchworking). Many restrictions on the topology of T-hypersurfaces are known. It would be interesting to look at these restrictions from the point of view of tropical geometry and to study the topology of real tropical varieties. For example, the following question arises.
Question: What can be said about Betti numbers of a real tropical variety?
(contributed by Ilia Itenberg)
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