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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.
Contents
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 factor.
Analog in non-zero characteristic
The conjecture is false over a field of characteristic p: any inseparable polynomial f(Xp) satisfies the condition since all derivatives are zero. Another, separable, counterexample is Xp+1 − Xp
Special cases
The conjecture is known to hold in characteristic zero for degrees which are prime power or twice a prime power: hence for all d up to 11. It has recently been established for d = 12.