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Let $a, b, d$ be real numbers and $c$ a complex number and define the quartic polynomial $$ z^4+z^3 (a+ i b)+z^2c+z (d-ib)+1 $$ I want to know sufficient and necessary conditions for when three of the roots are strictly inside the unit circle $|z| < 1$. By the fundamental theorem of algebra, the product of the roots is one, so there are at most three solutions strictly inside the unit circle. I'd also be happy if I could determine when there are three . Of course there exists an exact formula for determining the roots of a quartic polynomial, but it is extremely messy and doesn't give you the norm of the rooms.

Thanks!

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I think you can try to use Rouchè Theorem in the unitary circle...

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Let $A = a+ib$ and $B = d-ib$.
Given any $a,b,d$, the closed parametric curve $$ \gamma(t) = - e^{2it} -A e^{it} - B e^{-it} - e^{-2it}, \ 0 \le t \le 2 \pi $$ gives those $c$ for which there is a root on the unit circle. For $c$ outside the curve, by Rouché there are as many roots inside the unit circle as for $z^2$, namely two. As you cross the curve (at a regular point), the number either increases or decreases by $1$. If there are any $c$ that make the number $3$, they will form one or more of the regions into which the curve divides the complex plane.

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  • $\begingroup$ Thanks for your help! Just a quick question, if $c$ is outside the curve, how can Rouché's theorem be applied? Thanks! $\endgroup$
    – PhysKid
    Jun 6 '17 at 17:38
  • $\begingroup$ Rouché applies when $|c|$ is sufficiently large, as the term $c z^2$ dominates the others. And then the number of roots inside the circle stays constant in the unbounded component of the complement of the curve. $\endgroup$ Jun 6 '17 at 19:51
  • $\begingroup$ Thanks! Can I also use Rouché to argue that if my polynomial has three roots in the unit circle, then $|A|>|B|$? Because then the polynomial $z^3 A+z B$ has three roots inside the unit circle. Thus my sufficient and necessary conditions are that $c$ is inside the parametric curve you give and $|A|>|B|$? $\endgroup$
    – PhysKid
    Jun 6 '17 at 21:39
  • $\begingroup$ No, Rouché gives a sufficient condition, not a necessary one. $\endgroup$ Jun 6 '17 at 23:05
  • $\begingroup$ Could I not use the symmetric version of Rouché if $c$ is not large enough? Because based on my numerical trials in Mathematica, these are necessary and sufficient. $\endgroup$
    – PhysKid
    Jun 6 '17 at 23:15

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