Short & Sweet Math Challenge #21: Powers that be

[Paul Dale]

(11-08-2016 02:03 PM)J-F Garnier Wrote:  

(11-08-2016 01:38 PM)Paul Dale Wrote:  If there is one real root > 1 and all of the remaining (possibly complex) roots have |root| < 1, then the polynomial is of the required form.

It may be the element I was missing, but can you explain or justify this statement?
J-F

I'll try to justify the statement.

Newton's identities allow one to calculate the sum of a power of the roots of a polynomial from its coefficients. The constraints on the polynomials given here (monic with integral coefficients) mean that the sum of the roots to any power will be an integer.

For one root to approximate an integer as the power is raised, it must be the single dominant root. i.e. all other roots must approach zero as they are raise to higher and higher powers. Thus, their absolute value must be strictly less than unity. The dominant root must be one or greater.


Pauli