  For $n \times n$ matrices $A$ and $B$ define
$$\eta(A,B)=\sum_{S}\det(A[S])\det(B[S']),$$ where the summation is over all
subsets of $\{1,..., n\}$, $S'$ is the complement of $S$, and $A[S]$ is the
principal submatrix of $A$ with rows and columns indexed by $S$. We prove that
if $A\geq 0$ and $B$ is Hermitian then (1) the polynomial $\eta(zA,-B)$ has all
real roots (2) the latter polynomial has as many positive, negative and zero
roots (counting multiplicities) as suggested by the inertia of $B$ if $A>0$ and
(3) for $1\le i\le n$ the roots of $\eta(zA[\{i\}'],-B[\{i\}'])$ interlace
those of $\eta(zA,-B)$. Assertions (1)-(3) solve three important conjectures
proposed by C. R. Johnson 20 years ago. Moreover, we substantially extend these
results to tuples of matrix pencils and real stable polynomials. In the process
we establish unimodality properties in the sense of majorization for the
coefficients of homogeneous real stable polynomials and as an application we
derive similar properties for symmetrized Fischer products of positive definite
matrices. We also obtain Laguerre type inequalities for characteristic
polynomials of principal submatrices of arbitrary Hermitian matrices that
considerably generalize a certain subset of the Hadamard-Fischer-Koteljanskii
inequalities for principal minors of positive definite matrices. Finally, we
propose Lax type problems for real stable polynomials and mixed determinants.
