
Contents
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The ring K[∂] The ring K[∂]
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Right-inverses of linear differential operators Right-inverses of linear differential operators
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COROLLARY 5.1.5. The ring K[∂] has no ideals except f0g and K[∂]. COROLLARY 5.1.5. The ring K[∂] has no ideals except f0g and K[∂].
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Euclidean division in K[∂] Euclidean division in K[∂]
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The kernel of a linear differential operator The kernel of a linear differential operator
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Linear differential equations with constant coefficients Linear differential equations with constant coefficients
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The type of a linear differential operator The type of a linear differential operator
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Factorization into irreducibles and composition series Factorization into irreducibles and composition series
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Linear closedness and linear surjectivity Linear closedness and linear surjectivity
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5.2 Second-Order Linear Differential Operators 5.2 Second-Order Linear Differential Operators
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Application to linear differential operators with constant coefficients Application to linear differential operators with constant coefficients
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5.3 DIAGONALIZATION OF MATRICES 5.3 DIAGONALIZATION OF MATRICES
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Euclidean rings Euclidean rings
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Ore domains Ore domains
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Independence and rank Independence and rank
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Notes and comments Notes and comments
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Inhomogeneous equations Inhomogeneous equations
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Independence and finite-dimensionality Independence and finite-dimensionality
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Homogeneous equations Homogeneous equations
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DEFINITION 5.5.1. A differential module over K is a finite-dimensional vector space M over K together with a ∂-compatible derivation on M; we construe such M as a (left) K[∂]-module as indicated in the remarks preceding this definition. The dimension of a differential module M over K is the dimension dimK M of M as a vector space over K. DEFINITION 5.5.1. A differential module over K is a finite-dimensional vector space M over K together with a ∂-compatible derivation on M; we construe such M as a (left) K[∂]-module as indicated in the remarks preceding this definition. The dimension of a differential module M over K is the dimension dimK M of M as a vector space over K.
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LEMMA 5.5.5. e0; : : : ; en–1 are horizontal iff are horizontal. LEMMA 5.5.5. e0; : : : ; en–1 are horizontal iff are horizontal.
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Notes and comments Notes and comments
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Linear surjectivity in the presence of a valuation Linear surjectivity in the presence of a valuation
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Notes and comments Notes and comments
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5.7 COMPOSITIONAL CONJUGATION 5.7 COMPOSITIONAL CONJUGATION
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Transformation formulas Transformation formulas
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Compositional conjugation and upward shift Compositional conjugation and upward shift
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Compositional conjugation in the algebraic closure of KhY i Compositional conjugation in the algebraic closure of KhY i
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Compositional conjugation and rational powers Compositional conjugation and rational powers
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Notes and comments Notes and comments
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5.8 The Riccati Transform 5.8 The Riccati Transform
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The Riccati transform of a linear differential operator The Riccati transform of a linear differential operator
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Riccati transforms in the presence of a valuation Riccati transforms in the presence of a valuation
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5.9 Johnson’s Theorem 5.9 Johnson’s Theorem
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Tensor products of K[∂]-modules Tensor products of K[∂]-modules
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The module of differentials as a K[∂]-module The module of differentials as a K[∂]-module
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Modules of differentials for differential field extensions Modules of differentials for differential field extensions
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Independence at a prime Independence at a prime
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An abstract version of Johnson’s theorem An abstract version of Johnson’s theorem
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Notes and comments Notes and comments
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Five Linear Differential Polynomials
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Published:June 2017
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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.
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