*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

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 central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.Less

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 central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, *H*-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an *H*-field. This paves the way to a quantifier elimination with interesting consequences.

*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:
- 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

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 structure of a differential field. This introduction provides an overview of the importance of in different areas of mathematics such as analysis, computer algebra, and logic, as well as some of the most distinctive features of . In particular, it discusses the ordered and valued differential field , grid-based transseries, H-fields, closure properties, and valuations and asymptotic relations. It also explains the book's strategy and main results and considers a few more open-ended avenues of inquiry.Less

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 structure of a differential field. This introduction provides an overview of the importance of in different areas of mathematics such as analysis, computer algebra, and logic, as well as some of the most distinctive features of . In particular, it discusses the ordered and valued differential field , grid-based transseries, *H*-fields, closure properties, and valuations and asymptotic relations. It also explains the book's strategy and main results and considers a few more open-ended avenues of inquiry.