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Asymptotic Differential Algebra and Model Theory of Transseries

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

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.001.0001
Subject:
Mathematics, Computational Mathematics / Optimization

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a ... More


<i>H</i>-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/9780691175423.003.0011
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter considers H-fields, pre-differential-valued fields with a field ordering that interacts with the valuation and derivation. Axiomatizing this interaction yields the notion of a ... More


Introduction and Overview

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/9780691175423.003.0001
Subject:
Mathematics, Computational Mathematics / Optimization

This book develops the algebra and model theory of the differential field of transseries, which are formal series in an indeterminate x > ℝ. is a field containing ℝ as a subfield and acquires the ... More


Eventual Quantities, Immediate Extensions, and Special Cuts

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/9780691175423.003.0012
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with eventual quantities, immediate extensions, and special cuts. It first considers the behavior of eventual quantities before discussing Newton weight, Newton degree, and Newton ... More


The Newton Polynomial

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/9780691175423.003.0014
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter focuses on the Newton polynomial based on assumption that K is a differential-valued field of H-type with asymptotic integration and small derivation. Here K is also assumed to be ... More


Quantifier Elimination

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/9780691175423.003.0017
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter considers the theory Tsuperscript nl of ω‎-free newtonian Liouville closed H-fields that eliminates quantifiers in a certain natural language. This theory has two completions: in the ... More


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