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Let $$p(k)=a_0+a_1x+a_2x^2+a_3x ^3+\dots+a_kx^k$$ Is possible known just looking at the coefficients $a_0,a_1,\dots,a_k$ $(a_k\in\mathbb{R})$ if the polynomial $p(k)$ will have roots out of the unit circle for values of $k=2,3,4$?

EDIT: I'm asking about both complex and real roots and I want to know if there is a way to check if all the roots will be outside of the unit circle or in somehow verify that at least one of the roots will be inside the unit circle.

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    $\begingroup$ You may search for this paper: Palindrome-Polynomials with Roots on the Unit Circle by John Konvalina and Valentin Matache. $\endgroup$ – mike Sep 5 '17 at 19:48
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    $\begingroup$ It depends on what "just looking at" means. If there are no roots on the unit circle, the winding number of $a_0 + a_1 e^{i\theta} + \ldots + a_k e^{ik\theta}$ about the origin as $\theta$ goes from $0$ to $2\pi$ gives you the number of roots (counted by multiplicity) inside the unit circle. Subtract from $k$ (assuming $a_k \ne 0$) to get the number outside the circle. $\endgroup$ – Robert Israel Sep 5 '17 at 20:00
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    $\begingroup$ If the coefficients are positive (or alternate in sign) then you could see if the Kakeya-Enestrom theorem could be applied. $\endgroup$ – Antonio Vargas Sep 7 '17 at 15:02
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Firstly, to clarify - are you allowing complex roots or only real roots?

I do not know of an if or only if test. However, there are a couple of things that will help in some circumstances. Divide through by $a_k$ then:

  • $a_0$ is the product of the roots, so if it is outside the unit circle, then there must be a root also outside.
  • $a_1$ is the sum of the roots, so if its absolute is greater than k, then there must be a root outside the unit circle.

If this is a real life problem, then plugging your polynomial into a computer program is the way to go. If this is a question for mathematical interest, then I would be very interested to hear if someone knows a solution!

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