TY - GEN A1 - Gatermann, Karin T1 - Counting stable solutions of sparse polynomial systems in chemistry N2 - The polynomial differential system modelling the behavior of a chemical reaction is given by graphtheoretic structures. The concepts from toric geometry are applied to study the steady states and stable steady states. Deformed toric varieties give some insight and enable graph theoretic interpretations. The importance of the circuits in the directed graph are emphazised. The counting of positive solutions of a sparse polynomial system by B.\ Sturmfels is generalized to the counting of stable positive solutions in case of a polynomial differential equation. The generalization is based on a method by sparse resultants to detect whether a system may have a Hopf bifurcation. Special examples from chemistry are used to illustrate the theoretical results. T3 - ZIB-Report - 00-32 KW - mass action kinetics KW - toric variety KW - stability KW - circuit Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6007 ER -