abstract-algebrapolynomialsreal-analysis
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?
The Schur-Cohn algorithm and the Jury stability criterion answer some of your questions. Specifically, the Schur-Cohn algorithm decides, whether there are any roots inside the (closed) unit circle and the Jury stability test decides whether all the roots are strictly inside the unit circle.
This is not a answer. I just want to use the space and easy-to-edit feature in the "Answer" section to type the equations.
You did not specify the relation of the modulus of the coefficients $a_n$. If they happen to be cup-shaped, then you may use a theorem by Chen (J. of Math. Anal. and Appl. vol 190, 714-724 (1995)).
By cup-shaped, I mean
$$|a_0|\ge |a_1| \ge \cdots \le |a_{N-2}| \le |a_{N-1}|$$
In your case, we have (assume $N=2n+2$) $$P(z) = \sum_{k=0}^{2n+1}a_kz^k=z^n q(z)+q^*(z)$$
where $$q_n(z)=\sum_{k=0}^na_{n+k+1}z^k$$ $$q_n^*(z)=\sum_{k=0}^n a_kz^k$$
If $N=2n+1$, then we define $Q(z)=(1+z)P(z)$ and treat $Q(z)$ in a similar fashion.
Best Answer
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).