On the number of real polynomials of the Casas-Alvero type

Let K be a field and P ∈ K[X] is a polynomial of degree n, then the conjecture of Casas-Alvero states that if P is not prime with each of its n − 1 first derivatives, then it is a monomial, i.e., of the form c(X − r)n. We consider the case where K=ℝ and P   is split over ℝ, where we show that the number un of hypothetical counterexamples of degree n satisfies (n − 4) ! ≤ un ≤ c(n − 3)n−2, where c=2e−1(∏n=2∞e−1(∑k=0n1/k!))2=0.59373381…. We also show how the Rolle theorem implies simply some previous results (see [1], [2], [3] and [4]) and we improve them.