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

Univariate polynomials are called stable with respect to a circular region A, if all of their roots are in A. We consider the special case where A is a half-plane and investigate affine slices of the set of stable polynomials. In this setup, we show that an affine slice of codimension k always contains a stable polynomial that possesses at most 2 (k+2) distinct roots on the boundary and at most (k+2) distinct roots in the interior of A. This result also extends to affine slices of weakly Hurwitz polynomials, i. e. real, univariate, left half-plane stable polynomials. Subsequently, we apply these results to symmetric polynomials and varieties. Here we show that a variety described by polynomials in few multiaffine polynomials has no root in Aⁿ, if and only if it has no root in Aⁿ with few distinct coordinates. This is at the same time a generalization of the degree principle to stable polynomials and a generalization of Grace-Walsh-Szego's coincidence theorem.