Oskar Henriksson: Tropical root bounds: case studies from chemistry, rigidity and physics

Abstract.

Accurate upper bounds on the number of roots of a polynomial system play a key role in numerical algebraic geometry. In this work, we develop a tropical root-counting strategy for sharpening the classical BKK bound for parametrized polynomial systems with dependencies among the coefficients, which also gives an algorithm for constructing generalized polyhedral homotopies. In this talk, I will showcase three types of systems of applied interest where our tropical root bound is generically sharp: steady-state systems in chemistry, constraint equations in rigidity theory, and CHY scattering equations in particle physics. This is based on joint works with Elisenda Feliu, Paul Helminck, Yue Ren, Benjamin Schröter and Máté Telek.