The Wonderful Geometry of the Vandermonde map
Appears in collection : 2023 - T3 - WS2 - Geometry of polynomial system solving, optimization and topology
The main object of this talk is the so called Vandermonde map - the map given by a selection of powersum polynomials - which appears quite naturally in various contexts and thus providing connections between different mathematical domains. Our interest in this object is motivated by the following problem: Suppose that we are given a polynomial expression in traces of powers of symmetric matrices is there an algorithm to decide whether this expression is nonnegative for all symmetric matrices of all sizes? What happens if we replace trace by normalized trace? As one of the results of our work we show that the first (unnormalized) problem is undecidable, while the second one is decidable. The key to the hardness of the unnormalized problem is the fascinating geometry of the image of the probability simplex under the Vandermonde map.
