Toric geometry and dynamics of chemical reactions

Karin Gatermann (Berlin, Germany)

The theory of toric varieties is very important for investigating the structure of solutions of a sparse polynomial system. In this talk I will show that several ideas from toric geometry carry over to ordinary differential equations which are sparse polynomial as in the case of chemical reactions. The torus group action give rise to equivalence classes of differential equations. The lattice helps to find Lyapunov functions. Toric varieties are involved in determining stability of equilibria and Hopf bifurcation points. Most importantly, the rate constants of the chemical reactions play the role of deforming the toric variety into products of several smaller toric varieties. This gives some insight into the structure of positive solutions.