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Local tropical linear spaces. (English) Zbl 1281.14049
Tropical linear spaces were introduced by D. Speyer [SIAM J. Discrete Math. 22, No. 4, 1527–1558 (2008; Zbl 1191.14076)]. They are $$m$$-dimensional polyhedral complexes in $$\mathbb{R}^n$$, closed under sum of multiples of the vector $$(1,\ldots,1)$$. In this paper the author studies tropical linear spaces locally. More precisely, to every tropical linear space $$L$$ there corresponds a matroid $$M$$. Then to each basis $$B$$ of $$M$$, there corresponds a subcomplex $$L_B$$ of $$L$$, which is the local tropical linear space determined by $$B$$, by definition. Clearly $L=\bigcup_BL_B.$ If $$L$$ is $$m$$-dimensional, the author shows that $$L_B$$ is homeomorphic to Euclidean space $$\mathbb{R}^m$$, via a piecewise linear map. He gives a projection map from $$\mathbb{R}^n$$ onto any given tropical linear space $$L$$. On the combinatorial side, the author proves that local tropical linear spaces $$L_B$$ are dual to mixed subdivisions of Minkowski sums of simplices. This allows him to give upper bounds for the $$f$$-vector of $$L_B$$, making a distinction between the number of $$i$$-dimensional faces that become bounded when modding out by the lineality space spanned by the vector $$(1,\ldots,1)$$ and the number of arbitrary $$i$$-dimensional faces. Tight upper bound for these numbers are ${{n-i-1}\choose {n-m-i,i-1,m-i}}={{n-i-1}\choose {i-1}}{{n-2i}\choose {m-i}}$ and ${{n-i-1}\choose {m-i}}{{n-1}\choose {i-1}},$ respectively. These bounds are related to similar bounds for tropical linear spaces, conjectured by Speyer in 2008, and proved in some cases by the latter. This is the so called $$f$$-conjecture. The author finishes this very interesting and well-written paper by defining conical tropical linear spaces and giving a direct proof of the $$f$$-conjecture for them, based on the tight upper bounds he has found previously.

##### MSC:
 14T05 Tropical geometry (MSC2010) 52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
TropLi
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##### References:
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