Stable and Hurwitz slices, a degree principle and a generalized Grace-Walsh-Szegő theorem

We consider affine varieties defined by multivariate complex symmetric polynomials and address the question whether such a variety contains a point with only few distinct real and imaginary coordinates. We present a degree principle for half-planes and a weak generalization of Grace-Walsh-Szegő’s coincidence theorem to symmetric polynomials that are not necessarily multiaffine but which can be expressed in few elementary symmetric polynomials. The results are derived from studying linear slices of the set of univariate stable polynomials with respect to a closed half-plane using Vieta’s formulas. Similarly, we study affine slices of the set of weakly Hurwitz polynomials.

This is joint work with Cordian Riener and Robin Schabert