On the Casas-Alvero conjecture

The Casas–Alvero conjecture says that a complex univariate polynomial having roots in common with each of its derivatives must be a power of a linear polynomial. In this expository note we review some ideas on this conjecture. In particular, we explain the rather successful algebraic attack by Graf von Bothmer et al., which proves the conjecture in infinitely many degrees, and challenge analysts to come up with new approaches.