# A hybrid approach to some stability problems for bivariate polynomials

Frequently, in control system design, we are asked to locate the roots of a bivariate
polynomial
of the following form
\begin{eqnarray*}
H(s,k)=\sum_{i=0}^n Q_i(s) k^i\in {\bf Z}[k,s]={\bf Z}[k][s]={\bf D}[s],
\end{eqnarray*}
where $Q_i(s)\in {\bf Z}[s]$ for each $i$ and, moreover,
$k$ is a free parameter
ranging in some real interval.
For a fixed value $\bar k$ of $k$, the zero distribution
of the univariate polynomial $p(s)=H(s,\bar k)$
with respect
to the imaginary axis
can be found by determining the inertia of a
Bezout matrix $B=B(\bar k)$
whose entries
are expressed in terms of the coefficients of $p(s)$.
This evaluation is usually
accomplished by
computing a block factorization of $B$, namely,
$U^TBU=D$ where $D$ is a block diagonal matrix
with lower triangular blocks with respect to the antidiagonal.
It is intended in this paper to propose an efficient
hybrid approach for determining the zero-distribution of $H(s,k)$
with respect to the
imaginary axis for any value of $k$.
We develop a fast
fraction-free
method for factoring the Bezout matrix $B(k)$ with
entries over ${\bf D}$ determined by
$H(s,k)\in {\bf D}[s]$. In this way,
we easily compute the sequence
$\{\phi_i(k)\}$ of the trailing principal minors
of $B(k)$. For almost any value $\bar k$ of
$k$ the associated
sign sequence $\{sign(\phi_i(\bar k))\}$ specifies
the inertia of $B(\bar k)$
and, therefore, the zero-distribution of $H(s,\bar k)$.
The function
$sign(\phi_i( k))$ is finally obtained
by numerically computing rational approximations of the
real zeros of $\phi_i(k)\in {\bf Z}[k]$.