Casas-Alvero conjecture

The Casas-Alvero conjecture states that a complex polynomial sharing a root with each of its nonconstant derivatives must be a power of a linear polynomial.

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Polynomial derivative: The derivative of a polynomial is obtained by differentiating each power term and measures its rate of change. The conjecture imposes a common-root condition between the polynomial and every nonconstant derivative.
Joseph-Louis Lagrange: An Italian-French mathematician whose work shaped analysis, number theory, and mechanics. Lagrange posed an early form of the problem concerning shared roots with derivatives.
Fundamental theorem of algebra: The theorem that every nonconstant complex polynomial has at least one complex root. It guarantees a root for the polynomial, but not that the same root works for all derivatives.
Rolle's theorem: A theorem stating that a differentiable real function with equal values at two endpoints has a stationary point between them. It links roots of a real function to derivative roots, unlike the conjecture's shared-root requirement.
Resultant: A polynomial expression in the coefficients of two polynomials that vanishes exactly when they share a root over an algebraic closure. Resultants encode each required common root as an algebraic equation in the polynomial's coefficients.
Casas-Alvero problem: The problem asking whether a complex polynomial sharing a root with each nonconstant derivative must be a pure power. This named formulation is the historical source of the conjecture's title.
Casas-Alvero conjecture in degree two: The degree-two case of the Casas-Alvero conjecture, in which a shared root with the first derivative forces a double root. The derivative is linear, so its unique root must also be a root of the quadratic.
Gauss-Lucas theorem: A theorem stating that the roots of a polynomial's derivative lie in the convex hull of the polynomial's roots. It constrains derivative roots geometrically without requiring them to coincide with a root of the polynomial.
Discriminant: A polynomial expression in a polynomial's coefficients that vanishes when the polynomial has a repeated root. The first derivative condition forces a repeated root, detectable through the discriminant.
Eduardo Casas-Alvero: A Spanish mathematician known for work in algebraic geometry and for formulating the conjecture bearing his name. His treatment brought the problem into its modern conjectural form.