Hilbert's Irreducibility Theorem for Linear Differential Operators

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