Casas-Alvero conjecture

Let f be a polynomial of degree d defined over a field K of characteristic zero.

If f has a factor in common with each of its derivatives f(i), i = 1, ..., d − 1, then the conjecture predicts that f must be a power of a linear polynomial.

The conjecture is false over a field of characteristic p: any inseparable polynomial f(Xp) without constant term satisfies the condition since all derivatives are zero.

The conjecture is known to hold in characteristic zero for degrees of the form pk or 2pk where p is prime and k is a positive integer.

It has recently been established for d = 12, making d = 20 the smallest open degree.