Casas-Alvero conjecture

In mathematics, the Casas-Alvero conjecture is an open problem about polynomials which have factors in common with their derivatives, proposed by Eduardo Casas-Alvero in 2001.

Casas-Alvero talking about the conjecture in a conference in the University of Barcelona on March 16, 2016.

Formal statement

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.

Analogue in non-zero characteristic

The conjecture is false over a field of positive characteristic p: any inseparable polynomial f(X^p) without constant term satisfies the condition since all derivatives are zero. Another counterexample (which is separable) is X^p+1 − X^p.

Special cases

The conjecture is known to hold in characteristic zero for degrees of the form p^k or 2p^k where p is prime and k is a positive integer. Similarly, it is known for degrees of the form 3p^k where p ≠ 2, for degrees of the form 4p^k where p ≠ 3, 5, 7, and for degrees of the form 5p^k where p ≠ 2, 3, 7, 11, 131, 193, 599, 3541, 8009. Similar results are available for degrees of the form 6p^k and 7p^k. It has recently been established for d = 12, making d = 20 the smallest open degree.