The concept of invariant sets is an important tool to investigate the behaviour of solutions of ODE systems. A set $M$ is called invariant for a specified system if every trajectory of the system that starts in $M$ stays in $M$ for all times. Considering varieties and polynomial ODE systems, we show how Groebner bases provide an algorithmic test for invariance. As an application, we...
Motivated by singularity theory, Hiroaki Terao introduced a module of logarithmic derivations associated with a hyperplane arrangement. This talk is concerned with Terao's freeness conjecture which asserts that the freeness of this derivation module is determined by the underlying combinatorics of the arrangement.
To investigate this conjecture, we have enumerated all matroids of rank 3...
We introduce the integral matrix similarity problem as a special instance of the conjugacy problem from group theory. While this problem has been known to be decidable for over 40 years due to work of Grunewald and Sarkisyan, no upper bound on the complexity has been known and practical algorithms have only been available for special cases. We will present recent work with Werner Bley and...
Countless applied problems in various sciences can be expressed as polynomial optimization problems. Solving these nonlinear problems essentially requires to certify nonnegativity of multivariate, real polynomials, a classical problem from real algebraic geometry.
A classical way to certify nonnegativity are sums of squares (SOS). An alternative way are sums of nonnegative circuit polynomials...
It is very common nowadays to use tools from symbolic computation for applications in many different areas of mathematics (or physics, or computer science, etc.). One backbone of symbolic computation is Cylindrical Algebraic Decomposition (CAD). In the past years, we have applied CAD to problems arising in applied mathematics. Among the prominent methods in computer algebra are algorithms to...
