I have the following quartic equation $$\omega_4 x^4+\omega_3 x^3+\omega_2 x^2+\omega_1 x+\omega_0=0$$ where $\omega_i$ depend on several real parameters. I'm not interested in searching its roots, but I would like to respond to the following point:

  1. What are the bounds (lower and upper) for the absolute value of all the roots (real or complex)?

There is some radius $r$ such that when $|x|>r,$

$$|\omega_4 x^4| > |\omega_3 x^3| + |\omega_2 x^2| + |\omega_1 x| + |\omega_0| > |\omega_3 x^3 + \omega_2 x^2 + \omega_1 x + \omega_0| $$

Rouche's theorem says that when this is the case, all of the roots will be within a disk of radius $r.$

  • $\begingroup$ Thank you. What is $z$? If all the roots are real, this means that they will be in the range $[-x,x]$? $\endgroup$ – Mark Aug 17 '18 at 7:09
  • $\begingroup$ This is a proof of complex analysis where polynomials are usually expressed in terms of $z.$ But, as I have not introduced z, I will change it. $\endgroup$ – Doug M Aug 17 '18 at 16:39

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