A characterization related to the equilibrium distribution associated with a polynomial structure

Let f be a probability density function on (a, b) C (0, infinity) and consider the class Cf of all probability density functions of the form Pf where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x -> P(X > x)/E(X) also belongs to Cf : this happens for instance when f(x) = Ce-¿x or f(x) = C(b-x) ¿-1. The present paper shows that actually they are the only possible two cases. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.