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Differential Polynomials

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0005
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their ... More


Linear Differential Polynomials

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0006
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic ... More


Asymptotic Fields and Asymptotic Couples

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0010
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with asymptotic differential fields and their asymptotic couples. Asymptotic fields include Rosenlicht's differential-valued fields and share many of their basic properties. A key ... More


Triangular Automorphisms

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0013
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter focuses on triangular automorphisms, which can be analyzed by Lie techniques. Throughout the discussion K is a commutative ring containing ℚ as a subring. A formalism is introduced to ... More


Newtonian Differential Fields

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0015
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with Newtonian differential fields. Here K is an ungrounded H-asymptotic field with Γ‎ := v(Ksuperscript x ) not equal to {0}. So the subset ψ‎ of Γ‎ is nonempty and has no largest ... More


Newtonianity of Directed Unions

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0016
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter considers the newtonianity of directed unions and proves an analogue of Hensel's Lemma for ω‎-free differential-valued fields of H-type: Theorem 15.0.1. Here K is an H-asymptotic field ... More


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