Casas–Alvero conjecture: let \(f\) be a degree-\(d\) univariate polynomial over a field of characteristic zero sharing a (nonconstant) factor with each of its derivatives \(f', f'', \dots, f^{(d-1)}\). Then \(f\) is a power of a linear polynomial, \(f = a(x-b)^d\).
Proposed by Eduardo Casas-Alvero in 2001 from the study of higher-order polar germs of plane curve singularities. It is settled for degrees \(p^k\) and \(2p^k\) (von Bothmer–Labs–Schicho–van de Woestijne 2007) and several further families, with computer verification in low degrees — yet the general case resists, and claimed proofs have been withdrawn. The characteristic-zero hypothesis is essential: inseparable polynomials in characteristic \(p\) are counterexamples.
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