A b s t r ac t . A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians.2010 Mathematics Subject Classification. 52B40 (05B35, 14T05).

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A b s t r ac t . We study the fan structure of Dressians Dr (d, n) and local Dressians Dr(M) for a given matroid M. In particular we show that the fan structure on Dr(M) given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope P (M). As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse. arXiv:1809.08965v1 [math.CO] 24 Sep 2018 Cambridge University Press, Cambridge, 1986.I n s t i t u t f ü r M at h e m at i k , F U B e r l i n , A r n i m a l l e e 2 , 1 4 1 9 5 B e r l i n , G e r m a n y , E -m a i l : o l a rt e @ z e dat . f u -b e r l i n . d e I n s t i t u t f ü r M at h e m at i k , T U B e r l i n , S t r . d e s 1 7 . J u n i 1 3 6 , 1 0 6 2 3 B e r l i n , G e r m a n y , E -m a i l : pa n i z z u t @ m at h . t u -b e r l i n . d e D e pa rt m e n t o f M at h e m at i c a l S c i e n c e s , B i n g h a m t o n U n i v e r s i t y , B i n g h a m t o n , N Y 1 3 9 0 2 , U S A , E -m a i l : s c h ro e t e r @ m at h . b i n g h a m t o n . e d u

A b s t r ac t . We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application f -vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner and tropical bases.M. Joswig is supported by Einstein Foundation Berlin and Deutsche Forschungsgemeinschaft (DFG).

We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry.