@misc{Gatermann1990, author = {Gatermann, Karin}, title = {Gruppentheoretische Konstruktion von symmetrischen Kubaturformeln.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4652}, number = {TR-90-01}, year = {1990}, abstract = {\$G\$-invariant cubature formulas for numerical integration over n-dimensional, \$G\$- invariant integration regions are computed symbolically. The nodes are the common zeros of some \$d\$-orthogonal polynomials which build an \$H\$-basis of an ideal. Approaches for these polynomials depending on parameters are made with the help of the theory of linear representations of a group \$G\$. This theory is also used for the effective computation of necessary conditions which determines the parameters. Another approach uses invariant theory and gr{\"o}bner bases.}, language = {de} } @phdthesis{Gatermann1999, author = {Gatermann, Karin}, title = {Computer Algebra Methods for Equivariant Dynamical Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4140}, number = {SC-99-26}, year = {1999}, abstract = {An introductory chapter on Groebner bases is given which also includes new results on the detection of Groebner bases for sparse polynomial systems. Algorithms for the computation of invariants and equivariants for finite groups, compact Lie groups and algebraic groups are presented and efficient implementation and time comparision are discussed. This chapter also inlcudes improvements of the computation of Noether normalisation and Stanley decomposition. These results are applied in symmetric bifurcation theory and equivariant dynamics. As preparation of the investigation of the orbit space reduction three methods are compared for solving symmetric polynomial systems exactly. The method of orbit space reduction is improved by using the Cohen-Macaulayness of the invariant ring and nested Noether normalization. Finally this is applied for a case of mode interaction in the Taylor-Couette problem.}, language = {en} } @misc{GatermannHuber1999, 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}, year = {1999}, 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{Gatermann2000, 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}, year = {2000}, 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} }