I'm not sure if you can help me with this, but I'm currently looking for an upper bound on the real part of the roots of a polynomial with real coefficients. In other words, I have a polynomial

$a_n x^n + a_{n - 1} x^{n - 1} + ... + a_1 x + a_0 = 0$

where $a_0, a_1, \dots, a_{n}$ are real coefficients. Suppose that the roots are $t_1, t_2, \dots, t_n$. I want to find an bound $t^*$ such that $Re(t_i) \leq t^*$ for $i = 1, \dots, n$. Is there a theorem that provides such a result? I've seen Rouche's theorem, but from what I've read, this only tells me if the root lies in some circle of specified size. I may be missing something but I don't think that it tells me much about the real part of the root.

Any help is appreciated. Thank You.

  • $\begingroup$ IF you know that the root lies within a circle, you have bounded the root, and imagining the circle in the complex plane, it lies between two vertical lines (tangent to the circle) which provide explicit bounds and which can be determined as the real part of the circle +/- its radius. $\endgroup$ – Mark Bennet Jul 3 '11 at 20:15
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    $\begingroup$ That was supposed to be real part of the centre +/- radius $\endgroup$ – Mark Bennet Jul 3 '11 at 20:17
  • $\begingroup$ I can't imagine that knowing that the polynomial has real coefficients will help. $\endgroup$ – Thierry Zell Jul 3 '11 at 20:46
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    $\begingroup$ A real polynomial with all its roots in the left half of the complex plane is a Hurwitz stable polynomial. There are characterizations of the these polynomials, in particular the Routh-Hurwitz criterion. Or you can apply a M\"obius transformation to map the half-plane to a disc and apply the Schur-Cohn condition. $\endgroup$ – Chris Godsil Jul 3 '11 at 21:02
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    $\begingroup$ @Mark, @Chris, thank you for your comments. I should have mentioned this before, but the polynomial is convex and increasing on the right half plane and so has one positive real root. I do have much knowledge of the roots of polynomials, but does this imply that the real part of the complex roots lie in the left half plane (or is it still possible that the real part of these roots can be positive)? $\endgroup$ – user16149 Jul 3 '11 at 21:35

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