Research Group on Algebraic Geometry

From left to right: Martin Giese, Alex Zapletal, Josef Schicho, Tobias Beck, Niels Lubbes, Brian Moore, Janka Pilnikova, Jose Manuel Garcia Vallinas
(July 2006)

We develop mathematical theories, algorithms, and software for efficiently proving/disproving algebraic statements and solving algebraic constraints over the real numbers. The statements/conditions may contain inequalities and quantifiers. Many difficult problems in mathematics, scientific engineering and industrial computation can be reduced to that of solving algebraic constraints.

• Polynomial Equation Solving: We develop algorithms to get partial information about the solution set of a polynomial system of equations. E.g. we are looking for solutions in certain number fields.
• Parametrization: Some equational constraints can be solved by giving an "algebraic" parametrization of the solution set, i.e. a parametrization by rational functions. We are developing algorithms for finding parametrizations of algebraic surfaces, and for simplifying given parametrizations.
• Singularity Analysis: A solution set of algebraic constraints does in general have singularities, which are obstacles for identifying its topology or for visualizing it. Resolution is a standard way to analyze these singularities. We are developing algorithms for singularity resolution and studying applications (e.g. for the parametrization problem).