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.
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<sup> (i)</sup>, i = 1, ..., d â 1, then the conjecture predicts that f must be a power of a linear polynomial.
The conjecture is false over a field of positive characteristic p: any inseparable polynomial f(X<sup>p</sup>) without constant term satisfies the condition since all derivatives are zero. Another counterexample (which is separable) is X<sup>p+1</sup> â X<sup>p</sup>.
The conjecture is known to hold in characteristic zero for degrees of the form p<sup>k</sup> or 2p<sup>k</sup> where p is prime and k is a positive integer. Similarly, it is known for degrees of the form 3p<sup>k</sup> where p â 2, for degrees of the form 4p<sup>k</sup> where p â 3, 5, 7, and for degrees of the form 5p<sup>k</sup> where p â 2, 3, 7, 11, 131, 193, 599, 3541, 8009. Similar results are available for degrees of the form 6p<sup>k</sup> and 7p<sup>k</sup>. It has recently been established for d = 12, making d = 20 the smallest open degree.