  The Casas-Alvero conjecture states: if a complex univariate polynomial has a
common root with each of its derivatives, then it has a unique root. We show
that hypothetical counterexamples must have at least 5 different roots. The
first case where the conjecture is not known is in degree 12. We study the case
of degree 12, and more generally degree p+1, where p is a prime number. While
we don't come closing to solving the conjecture in degree 12, we present
several further constraints that counterexamples would have to satisfy.
