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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.)
Software:
TropLi
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References:
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