  Univariate polynomials are called stable with respect to a circular region
$\mathcal{A}$, if all of their roots are in $\mathcal{A}$. We consider the
special case where $\mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathcal{A}^n$, if and only if it has
no root in $\mathcal{A}^n$ 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-Szeg\H{o}'s coincidence theorem.
