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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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Linear Differential Polynomials

Linear Differential Polynomials

Chapter:
(p.241) Chapter Five Linear Differential Polynomials
Source:
Asymptotic Differential Algebra and Model Theory of Transseries
Author(s):

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175423.003.0006

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 KK, 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.

Keywords:   linear differential polynomial, homogeneous differential polynomial, linear differential operator, differential field, linear differential equation, differential module, valuation, compositional conjugation, Riccati transform, Johnson's Theorem

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