Rolle’s Theorem For Complex Roots Of Polynomials Of Small Degree∗

There are polynomials of odd degree d, for which Rolle’s property does not hold, i.e., polynomials such that for some pair of roots their formal derivative has no root “in between ” them. We describe in detail a counter-example for degree d = 7 and show that we cannot adapt the method to get a counter-example when d = 5. This case is the smallest unknown case. However, we prove that all polynomials of degree 5 with real coefficients do have Rolle’s property.