Subject: Characterization of positive polynomials

Dear Professor,
            
             I am very grateful if the following mathematical problem is 
solved, and the related web site is found. 

Let n be a positive integer and p(x) is a real polynomial of degree 
n-1.  Find a necessary and sufficient condition for the coefficients
of p(x) such that p(x)>=0 on the closed interval -1<=x<=1.  An
equivalent condition is the set of coefficients 

{(a_0, a_1, ..., a_{n-1}): p(1), p(-1), p(x_0) >=0 for all
turning points p'(x_0)=0}.

Is there any handy or parametric way of representing those 
coefficients?

Thanks a lot

Warmest regards,
Simon

Re: Characterization of positive polynomials