New Algorithms for Solving Tropical Linear Systems

The problem of solving tropical linear systems, a natural problem of tropical mathematics, has already proven to be very interesting from the algorithmic point of view: it is known to be in $NP\cap coNP$ but no polynomial time algorithm is known, although counterexamples for existing pseudopolynomial algorithms are (and have to be) very complex. In this work, we continue the study of algorithms for solving tropical linear systems. First, we present a new reformulation of Grigoriev's algorithm that brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us formulate a whole family of new algorithms, and we present algorithms from this family for which no known superpolynomial counterexamples work. Second, we present a family of algorithms for solving overdetermined tropical systems. We show that for weakly overdetermined systems, there are polynomial algorithms in this family. We also present a concrete algorithm from this family that can solve a tropical linear system defined by an $m\times n$ matrix with maximal element $M$ in time $\Theta\left({m \choose n} \mathrm{poly}\left(m, n, \log M\right)\right)$, and this time matches the complexity of the best of previously known algorithms for feasibility testing.Comment: 17 page

The Newton Polytope of the Implicit Equation

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure

Analysis of Biochemical Reaction Networks using Tropical and Polyhedral Geometry Methods

The field of systems biology makes an attempt to realise various biological functions and processes as the emergent properties of the underlying biochemical network model. The area of computational systems biology deals with the computational methods to compute such properties. In this context, the thesis primarily discusses novel computational methods to compute the emergent properties as well as to recognize the essence in complex network models. The computational methods described in the thesis are based on the computer algebra techniques, namely tropical geometry and extreme currents. Tropical geometry is based on ideas of dominance of monomials appearing in a system of differential equations, which are often used to describe the dynamics of the network model. In such differential equation based models, tropical geometry deals with identification of the metastable regimes, defined as low dimensional regions of the phase space close to which the dynamics is much slower compared to the rest of the phase space. The application of such properties in model reduction and symbolic dynamics are demonstrated in the network models obtained from a public database namely Biomodels. Extreme currents are limiting edges of the convex polyhedrons describing the admissible fluxes in biochemical networks, which are helpful to decompose a biochemical network into a set of irreducible pathways. The pathways are shown to be associated with given clinical outcomes thereby providing some mechanistic insights associated with the clinical phenotypes. Similar to the tropical geometry, the method based on extreme currents is evaluated on the network models derived from a public database namely KEGG. Therefore, this thesis makes an attempt to explain the emergent properties of the network model by determining extreme currents or metastable regimes. Additionally, their applicability in the real world network models are discussed

Tropical conics for the layman

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.Comment: 19 pages, 4 figures. Major rewriting of formerly entitled paper "Metric invariants of tropical conics and factorization of degree--two homogeneous tropical polynomials in three variables". To appear in Idempotent and tropical mathematics and problems of mathematical physics (vol. II), G. Litvinov, V. Maslov, S. Sergeev (eds.), Proceedings Workshop, Moscow, 200