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Fermat–Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing consensus trees in phylogenetics. This method has several desirable properties; e.g., it is Pareto and co-Pareto on rooted triplets.

We give a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. This yields lower bounds on the number of real roots of certain systems of polynomial equations known as “Wronski systems.”

We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application ff-vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner and tropical bases.

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are 344843867 such cones, organized into a database of 14373645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.

The$d$ -dimensional simplicial, terminal, and reflexive polytopes with at least$3d-2$vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves previous results of Casagrande (2006) and Øbro (2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics. This text is an extended abstract of Assarf et al. (2012). Nous classifions les polytopes simpliciaux, terminaux et réflexifs de dimension$d$avec au moins$3d-2$sommets. En particulier, tous ces polytopes se trouvent être des polytopes de Fano lisses. Nous améliorons des résultats antérieurs de Casagrande (2006) et d’Øbro (2008). Les polytopes de Fano lisses apparaissent en géométrie algébrique et en physique mathématique. Ce texte est un résumé étendu de Assarf et al. (2012).

The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes the phenotypes over many genetic loci, and the shape of this landscape can be used to predict evolutionary trajectories. Epistasis in a fitness landscape makes prediction of evolutionary trajectories more complex because the interactions between loci can produce local fitness peaks or troughs, which changes the likelihood of different paths. While various mathematical frameworks have been proposed to investigate properties of fitness landscapes, Beerenwinkel et al. (Stat Sin 17(4):1317–1342, 2007a) suggested studying regular subdivisions of convex polytopes. In this sense, each locus provides one dimension, so that the genotypes form a cube with the number of dimensions equal to the number of genetic loci considered. The fitness landscape is a height function on the coordinates of the cube. Here, we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool, which provides a concise combinatorial way of processing metric information from epistatic interactions. Furthermore, we extend the calculation of genetic interactions to consider interactions between microbial taxa in the gut microbiome of Drosophila fruit flies. We demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information on the fitness landscape where the previous approach is less conclusive.

Abstract

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on certain combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games. Mean payoff games are known to be in ${NP} \\cap {co-NP}$; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a nondegenerate linear program over Hahn series.