The Casas-Alvero conjecture concerns univariate polynomials and the roots they share with their own derivatives. In its usual form it asserts that if a complex univariate polynomial has a common root with each of its derivatives, then it has only one root.[1] An equivalent way of putting it is that such a polynomial must be a power of a linear polynomial, so that no genuinely non-trivial example can exist.[2] The problem was raised in 2001 by E. Casas-Alvero.[2] Stated over a general field, the conjecture predicts that a univariate polynomial over a field of characteristic zero sharing a common factor with each of its derivatives is of that restricted shape.[3]
The condition is easy to satisfy for a power of a linear polynomial and appears very hard to satisfy otherwise, which is what gives the problem its character. Much of the surrounding literature therefore studies necessary and sufficient conditions for a polynomial of given degree to be trivial in this sense.[4] The characteristic of the ground field matters, because the analogous statement can fail in positive characteristic for particular primes and degrees.[3] Over a field of positive characteristic the corresponding assertion is simply false.[6] In characteristic zero, by contrast, the conjecture has been confirmed for certain families of degrees.[6]
Constraints on any hypothetical counterexample are known. A counterexample must have at least five distinct roots.[1] The first degree in which the conjecture was not settled is 12.[1] That case has been examined directly, along with the more general family of degrees of the form p+1 for a prime p.[1] One recurring strategy is to establish the conjecture for polynomials of some small degree first and then propagate the result upward.[3] Because this propagation depends on the characteristic, attention falls on the primes for which the conjecture fails in a given degree and characteristic.[3] Explicit computations of distinguished monomials in the relevant resultant are used to identify such primes.[3] The conjecture also sits alongside a family of related questions about polynomials and their derivatives that were posed in the same setting.[4]
A separate line of attack replaces root conditions with geometry, studying varieties obtained by parameterizing polynomials of derivatives.[5] That approach draws on the Combinatorial Nullstellensatz and on Noether normalization.[5] It argues that the parameterizing polynomials form regular sequences.[5] From the resulting dimension count its authors announce a proof of the conjecture.[5] Announcements of this kind have appeared more than once, and the statement continues to attract both computational and theoretical work.[4]