Boolean networks can be viewed as functions on the set of binary strings of a given length, described via logical rules. They were introduced as dynamic models into biology, in particular as logical models of intracellular regulatory networks involving genes, proteins, and metabolites. Since genes can have several modes of action, depending on their expression levels, binary variables are...
In this talk, we develop a tropical analogue of eigenvalue methods in order to effectively compute the solution set of tropical polynomial systems. Relying on the connection between tropical linear systems and mean payoff games, we show that this solution set can be obtained by solving parametric mean-payoff games, arising from approriate linearizations of the tropical polynomial system...
We use a version Viro's patchworking theorem to construct systems with number of positive solutions which is at least the size of the largest positively decorable subcomplex of a regular triangulation of a polytope. We explain a duality result about such subcomplexes which enables us to get the currently best lower bounds known on the maximal number of positive solutions of polynomial systems...
Polyhedral homotopies were originally introduced by Huber and Sturmfels nearly 30 years ago, and have since become a staple strategy for solving polynomial systems. Main topic of the talk is a generalisation thereof. Building on ideas of Jensen, Leykin, and Yu, we will discuss two distinct types of tropical homotopies: First, we will discuss how to use tropical points to construct homotopies...
We develop a polyhedral geometry framework for dealing with higher rank valuations on function fields of algebraic varieties. As application, we explain the wall-crossing behavior of Newton-Okounkov bodies. Morever, using the set-up, we define combinatorial valuations on tropical varieties, and construct combinatorial Newton-Okounkov bodies. Based on joint work with Hernan Iriarte.
A tropical polynomial map is a piecewise-linear map between real Euclidean spaces. These maps represent a degeneration of classical polynomial maps between Euclidean spaces over valued fields. Accordingly, some of the pertinent classical topological invariants can be translated to polyhedral ones. In this talk, I will define the tropical analogue of the non-properness set of polynomial maps....
Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by particle physics. The main objects...
