Solving algebraic equations

The general goal is to develop tools and techniques for the construction and the geometric understanding of the solution variety of polynomial equations, especially such for which the solution set is expected to have singularities. Special emphasis is given to solutions in real affine space, in power series rings, and in p-adic spaces. Depending on which information we want to extract from the solution variety, we have various separate subgoals. Visual appearance: We wish to develop qualitative criteria that classify completely the geometric features of a real algebraic variety which determine its visual apperance (such as the link, the cones, vanishing cycles etc.); and to devise algorithms for checking these criteria. Arc spaces: The theory of power series solutions of algebraic equations is mostly existential. We will try to make it constructive as far as possible. More precisely, it is known that solving a system of equations modulo sufficiently high degree ensures to have a solution in power series. We intend to compute these bounds explicitly and and to develop efficient methods that compute power series solutions from that order on to an arbitrary order. It is also intended to generalize these methods to singular situations and to many variables. Fibrations: Algorithms shall be developed for detecting natural families of algebraic varieties generating a given variety. In addition to these theoretical and computational program want to use the established techniques in order to produce media for demonstrating algebraic varieties to an audience inside and outside mathematics.

The general goal is to develop tools and techniques for the construction and the geometric understanding of the solution variety of polynomial equations, especially such for which the solution set is expected to have singularities. Special emphasis is given to solutions in real affine space, in power series rings, and in p-adic spaces. Depending on which information we want to extract from the solution variety, we have various separate subgoals. Visual appearance: We wish to develop qualitative criteria that classify completely the geometric features of a real algebraic variety which determine its visual apperance (such as the link, the cones, vanishing cycles etc.); and to devise algorithms for checking these criteria. Arc spaces: The theory of power series solutions of algebraic equations is mostly existential. We will try to make it constructive as far as possible. More precisely, it is known that solving a system of equations modulo sufficiently high degree ensures to have a solution in power series. We intend to compute these bounds explicitly and and to develop efficient methods that compute power series solutions from that order on to an arbitrary order. It is also intended to generalize these methods to singular situations and to many variables. Fibrations: Algorithms shall be developed for detecting natural families of algebraic varieties generating a given variety. In addition to these theoretical and computational program want to use the established techniques in order to produce media for demonstrating algebraic varieties to an audience inside and outside mathematics.