Hilbert’s irreducibility theorem for linear differential operators

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We prove a differential analogue of Hilbert’s irreducibility theorem. Let L be a linear differential operator with coefficients in C ( X ) ( x ) that is irreducible over C ( X ) ¯ ( x ) , where X is an irreducible affine algebraic variety over an algebraically closed field C of characteristic zero. We show that the set of c ∈ X ( C ) such that the specialized operator L c of L remains irreducible over C ( x ) is Zariski dense in X ( C ) .