Mathematics Colloquia and Seminars

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We study a system of polynomial equations arising in modelling chemical reactions. The coefficients and the exponents of the polynomials are given by a weighted directed graph (some chemical reactions) and a weighted bipartite graph. For the study of stability of solutions special Liapunov functions are used. That means the optimization function is to be found which attains its minimum at the solution. I will explain the benefits of a certain convex polyhedral cone which comes together with a deformed toric variety. For the computation of the minimal generators of the cone one may of course use the simplex algorithm because under an additional constraint the cone gives the feasible set of a linear optimization problem. But in this particular situation many generators are given by positive circuits of the directed graph with consequences. In the second part of my talk I will generalize the counting of positive solutions of a sparse polynomial system to the counting of stable positive solutions.