@misc{Gatermann, author = {Gatermann, Karin}, title = {Counting stable solutions of sparse polynomial systems in chemistry}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6007}, number = {00-32}, abstract = {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.}, language = {en} } @misc{HuberRambauSantos, author = {Huber, Birkett and Rambau, J{\"o}rg and Santos, Francisco}, title = {The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3874}, number = {SC-98-44}, abstract = {In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of \emph{coherent} mixed subdivisions of a Minkowski sum \$\mathcal{A}_1+\cdots+\mathcal{A}_r\$ of point configurations and of \emph{coherent} polyhedral subdivisions of the associated Cayley embedding \$\mathcal{C}(\mathcal{A}_1,\dots,\mathcal{A}_r)\$. In this paper we extend this correspondence in a natural way to cover also \emph{non-coherent} subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.}, language = {en} } @misc{GatermannHuber, author = {Gatermann, Karin and Huber, Birkett}, title = {A family of sparse polynomial systems arising in chemical reaction systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4150}, number = {SC-99-27}, abstract = {A class of sparse polynomial systems is investigated which is defined by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results.}, language = {en} } @misc{DaisHausHenk, author = {Dais, Dimitrios I. and Haus, Utz-Uwe and Henk, Martin}, title = {On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3559}, number = {SC-98-12}, abstract = {\noindent An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces \$\Bbb{C}^{r}/G\$ in dimensions \$r\geq 4\$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.}, language = {en} }