# Do the roots of a polynomial remain inside the unit circle if we make the coefficients all positive?

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?

• Reminds me of Gershgorin circle theorem . 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).