Casas-Alvero conjecture
# 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.

## 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 characteristic p: any inseparable polynomial $f(X^p)$ without constant term satisfies the condition since all derivatives are zero. Another, separable, counterexample 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.

