132 Maxwell’s problem

A polynomial $h(x)$ is said to be stable if all of its zeros are in the open left half plane .

Given a monic polynomial, find necessary and sufficient conditions on the coefficients for the polynomial to be stable.

Finding the zeros exactly can be very difficult. Practical modern method: given a monic complex polynomial $p(s)$, there exists a complex matrix $A$ such that $det(sI-A)=p(s)$. Then one can find the eigenvalues of $A$ numerically. If all the eigenvalues are comfortably in the open left half plane, then $p(s)$ is stable. We now give a necessary condition for stability, which is not sufficient. Routh extended this to a sufficient condition.