MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
If you are asking about bounds on the roots (which the title would seem to indicate), then Fujiwara's bound might be useful:… ; however, I am not sure precisely what you are asking. – András Salamon Jun 6 '13 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 '13 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.