Polynomial convex coefficients

Question

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that the latter condition imposes some restrictions on $\theta_1,\theta_2,\theta_3,...$. Then my question is whether this conditions form a convex set.

For example if polynomial is of the order of 1 then

\begin{align*} & 1-L\theta =0 \\ & \rightarrow L=1/ \theta \\ & \rightarrow |1/ \theta|>1 \\ & \rightarrow |\theta|<1 \end{align*} the last equation forms a convex set. Now is that in general true?

Answer 1

The answer is that convexity doesn't hold for all $N$. For a polynomial with degree $N$, you may define a new polynomial $$ g(L) := L^N f(1/L). $$

We have that $f$ has all of its roots outside the unit circle if and only if $g$ has all of its roots inside the unit circle. This is also known in the literature as $g$ being Schur stable. Example 2.3 of [1] shows that the set of Schur stable polynomials for $N=5$ is not convex.

The problem of studying the set of the polynomials which are Schur stable seems to be well known in the literature, see for instance [2] and the references therein. In fact, the person who pointed out Example 2.3 for me was the corresponding author of [2].

[1] Shankar P. Bhattacharyya, H. Chapellat, and Lee H. Keel. Robust Control: The Parametric Approach. Upper Saddle River, New Jersey: Prentice-Hall, 1995.

[2] Baltazar Aguirre-Hernández, José Luis Cisneros-Molina, and Martín-Eduardo Frías-Armenta. Polynomials in Control Theory Parametrized by Their Roots. International Journal of Mathematics and Mathematical Sciences (2012).