On polynomial solutions of linear operator equations

Introduction Let K be a field of characteristic 0 and L : K[x] ! K[x] an endomorphism of the K-linear space of univariate polynomials over K. We consider the following computational tasks concerning L: T1. Homogeneous equation Ly = 0: Compute a basis of Ker L in K[x]. T2. Inhomogeneous equation Ly = f : Given f 2 K[x], compute a basis of the affine space L \Γ1 (f) in K[x]. T3. Parametric inhomogeneous equation Ly = P m i=1 i f i : Given f1