# Bounding Roots of a Polynomial by Coefficients

I'm using Samuelson's result and a chapter from Marden's monograph "The Geometry of Polynomials". These are sophisticated results. Are these independent from the Jury-Cohn test to show that a polynomial has roots less than unity?

For expository reasons (to get a better understanding) I'd like to use the results for functions with real coefficients of second and third degree and find the restrictions on the coefficients. Is this a more difficult task than setting a polynomial $a_2 x^2 + a_1 x + a_0 = 0$ and manipulating coefficients such that the roots $\lambda_1$ and $\lambda_2$ are less than 1? Is there a result on this somewhere already? Thank you

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If you are asking about bounds on the roots (which the title would seem to indicate), then Fujiwara's bound might be useful: en.wikipedia.org/wiki/… ; however, I am not sure precisely what you are asking. –  András Salamon Jun 6 at 12:23
Hi András, perhaps to make it more clear, do you know of a resource to fix the roots of the polynomial to be exactly modulus one, and then perturbations should should which direction the inequality would move? –  Jared Jul 2 at 22:02

For real $a,b,c$ in $ax^2+bx+c=0,$ you can get conditions on a,b,c explicitly in Mathematica using Resolve:

Resolve[ForAll[{x, y},
c + b x + a x^2 == 0 &&
b y + 2 a x y - a y^2 == 0  , (x^2 + y^2 < 1)], Reals]


Here, I used the substitution $z=x+iy$ and separated real and imaginary parts. This gives an ugly expression, but is fully automatic.

Looking at the expression, it seems very tedious to do it by hand.

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