Zusammenfassung für den Vortrag am 30.07.2021 (11:00 Uhr)

Seminar on Nonlinear Algebra

Christoph Schulze (TU Dresden)
Cones of locally non-negative polynomials

The study of non-negative polynomials is motivated by the obvious fact that the value at a global minimum of a real polynomial \(f\) is the maximal value \(c\) such that \(f-c\) is globally non-negative. This shows its connection to optimization. Similarly, a local minimum \(x_0\) of \(f\) induces the polynomial \(f-f(x_0)\) which takes value \(0\) and is locally non-negative at \(x_0\).

I will present results from my PhD thesis on the convex cone of locally non-negative polynomials. We will see geometric interpretations and examples of faces of this cone, some general theory of cones in infinite-dimensional vector spaces and classifications of faces using tools from singularity theory. I will also give a short outlook on an application to sums of squares in real formal power series rings.