Discriminants of parametric families of positive toric varieties
Binomial ideals are well established mathematical structures and they are often encountered in applications. Examples of recent applications include Algebraic Statistics and Algebraic Systems Biology. Geometrically a binomial ideal corresponds to the union of several toric varieties. Toric varieties are often described in terms of monomial parametrizations which makes them computationally tractable. In many applications only the nonnegative roots of a polynomial system have physical meaning. For binomial ideals this is especially interesting because at most one irreducible component of their varieties intersects the interior of the positive orthant.
The roots of polynomials often represent modi operandi of dynamical systems. Hence in principle the number of roots has a physical interpretation. It is often the case that measurement data for these systems is noisy, hence it is encoded in a set of parameters. In order to be consistent with this fact, it is crucial that the error bars do not intersect several regions in the space of parameters corresponding to different number of roots. Hence the classification of the parameter space with respect to the number of roots plays a core role. Such classification can be done through the discriminant variety. In the case of systems with positive roots, Sturm Discriminants can be systematically used to classify the parameters. As discriminantal methods heavily depend on elimination ideals, the existence of a binomial basis greatly reduces the computation time through the use of monomial parametrizations.
In this project we propose the use of Gröbner bases and Sturm discriminants for the study of parametric families of positive toric varieties, and their applications.