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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?

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  • $\begingroup$ Can you explain what's an "infinite order polynomial"? I have never heard of such thing. $\endgroup$ May 3, 2014 at 23:24
  • $\begingroup$ Dear @AlexandreEremenko. You mean it is not possible for a polynomial to have a very high order? $\endgroup$
    – TPArrow
    May 4, 2014 at 15:51
  • $\begingroup$ "Very high" or "infinite"? $\endgroup$ May 5, 2014 at 12:05
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    $\begingroup$ There is the Laguerre-Poyla class of functions that are limits of real stable polynomials.. $\endgroup$ Feb 13, 2016 at 16:26

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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).

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  • $\begingroup$ Thanks @Shamisen. I think if all the roots of f cite outside a unit circle, then, all the roots of g cite inside and on the border of unit circle. is that true? $\endgroup$
    – TPArrow
    May 8, 2014 at 13:28
  • $\begingroup$ @HAMEDHM a root of f site on the border of the unit circle if and only if a root of g site on the border of the unit circle. $\endgroup$
    – Tadashi
    May 8, 2014 at 13:50
  • $\begingroup$ between, if the answer satisfied you, you may accept this answer by clicking on the green check on the left side :) $\endgroup$
    – Tadashi
    May 8, 2014 at 13:51
  • $\begingroup$ Hello @HAMEDHM, thanks for the vote up! :) But I was meaning to accept the question as explained there: meta.stackexchange.com/questions/5234/… $\endgroup$
    – Tadashi
    May 8, 2014 at 14:25
  • $\begingroup$ Dear @Shamisen, could you please take a look at my question in mathoverflow.net/questions/168468/compact-and-convex-set $\endgroup$
    – TPArrow
    May 28, 2014 at 20:30

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