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?



Polynomial convex coefficients
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).


