Suppose the roots $r_1, \dots, r_p$ of the polynomial

$$x^p + a_1 x^{p-1} + a_2 x^{p-2} + \dots + a_p$$

all lie inside the unit circle. Is it true that the the roots of the polynomial

$$x^p + |a_1| x^{p-1} + |a_2| x^{p-2} + \dots + |a_p|$$

will also lie inside the unit circle?

  • $\begingroup$ Reminds me of Gershgorin circle theorem . $\endgroup$
    – lightxbulb
    Feb 7 '19 at 1:59

No. For instance, $$(x-1)^3(x+1)=x^4-2x^3+2x-1$$ has all its roots in the unit disk but $$x^4+2x^3+2x+1$$ does not (it has a root between $-3$ and $-2$). If you want the roots of the original polynomial to be in the open unit disk, you can perturb this example slightly (for instance, $(x-0.9)^3(x+0.9)$ still has a root near $-2$ when you take absolute values of its coefficients).

  • $\begingroup$ Thank you! Is it true up to polynomials of order 3? $\endgroup$
    – Monolite
    Feb 7 '19 at 11:46
  • $\begingroup$ I think so, though I haven't worked out a full proof. In low degrees you can learn quite a lot from the fact that the product of the roots is the constant term (up to sign) and the nonreal roots must come in conjugate pairs. $\endgroup$ Feb 7 '19 at 15:47

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