Re: Characterization of positive polynomials

Hi Simon,

The answer is almost. We have the following statement which appears as an 
exercise in Problems and Theorems in Analysis Vol.2 (page 78) by Polya and 
Szego,  that every polynomial of degree n which can assume only real and 
non-negative values for -1 <= x <= 1 may be represented in the form
        (A(x))^2 +(1-x^2)(B(x))^2, 
where A(x) and B(x) are polynomials of nth and (n-1)th degrees, 
respectively, with only real coefficients. 

Conversely and trivially, if we have a polynomial of the form A(x)^2 +(1-
x^2)B(x)^2, then it is nonnegative on [-1, 1]. The only thing it matters 
is that  it has to have the right degree, as this in general will be a 
polynomial of degree 2n instead of n. So if we start with a polynomial of 
degree n, we need to find constraints on A(x) and B(x) so that A(x)^2+(1-
x^2)B(x)^2 will be of degree n.  This will yield a system of equations on 
the coefficients of A(x) and B(x) which yields a parametric description. 
This is the messy part and that is why the answer is almost. 

You may want to have a look at the book mentioned above. It is a great 
book. 

Hope this will help.

Xiao