Linear Differential Polynomials

Abstract

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic results on them. In particular, it describes the property of a linear differential operator over a differential field K of defining a surjective map K → K, along with the transformation of a system of linear differential equations in several unknowns to an equivalent system of several linear differential equations in a single unknown. The chapter also discusses second-order linear differential operators, diagonalization of matrices, differential modules, linear differential operators in the presence of a valuation, and compositional conjugation. It concludes with an analysis of the Riccati transform and Johnson's Theorem.