*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/9780691175423.003.0012
- 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

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 multiplicity as well as Newton weight of linear differential operators. It then establishes the following result: Every asymptotically maximal H-asymptotic field with rational asymptotic integration is spherically complete. The chapter proceeds by describing special (definable) cuts in H-asymptotic fields K with asymptotic integration and introducing some key elementary properties of K, namely λ-freeness and ω-freeness, which indicate that these cuts are not realized in K. It shows that has these properties. Finally, it looks at certain special existentially definable subsets of Liouville closed H-fields K, along with the behavior of the functions ω and λ on these sets.Less

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 multiplicity as well as Newton weight of linear differential operators. It then establishes the following result: Every asymptotically maximal *H*-asymptotic field with rational asymptotic integration is spherically complete. The chapter proceeds by describing special (definable) cuts in *H*-asymptotic fields *K* with asymptotic integration and introducing some key elementary properties of *K*, namely λ-freeness and ω-freeness, which indicate that these cuts are not realized in *K*. It shows that has these properties. Finally, it looks at certain special existentially definable subsets of Liouville closed *H*-fields *K*, along with the behavior of the functions ω and λ on these sets.

*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/9780691175423.003.0015
- 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

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 element, and thus K is pre-differential-valued by Corollary 10.1.3. An extension of K means an H-asymptotic field extension of K. The chapter first considers the relation of Newtonian differential fields to differential-henselianity before discussing weak forms of newtonianity and differential polynomials of low complexity. It then proves newtonian versions of d-henselian results in Chapter 7, leading to the following analogue of Theorem 7.0.1: If K is λ-free and asymptotically d-algebraically maximal, then K is ω-free and newtonian. Finally, it describes unravelers and newtonization.Less

This chapter deals with Newtonian differential fields. Here *K* is an ungrounded *H*-asymptotic field with Γ := *v*(*K*superscript x ) not equal to {0}. So the subset ψ of Γ is nonempty and has no largest element, and thus *K* is pre-differential-valued by Corollary 10.1.3. An extension of *K* means an *H*-asymptotic field extension of *K*. The chapter first considers the relation of Newtonian differential fields to differential-henselianity before discussing weak forms of newtonianity and differential polynomials of low complexity. It then proves newtonian versions of d-henselian results in Chapter 7, leading to the following analogue of Theorem 7.0.1: If *K* is λ-free and asymptotically d-algebraically maximal, then *K* is ω-free and newtonian. Finally, it describes unravelers and newtonization.

*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/9780691175423.003.0016
- 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

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 with asymptotic couple (Γ, ψ), and γ ranges over Γ. The chapter first describes finitely many exceptional values, integration and the extension K(x), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If K is d-valued with ∂K = K, and K is a directed union of spherically complete grounded d-valued subfields, then K is newtonian. In concrete cases the hypothesis K = ∂K in the theorem can often be verified by means of Corollary 15.2.4.Less

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 with asymptotic couple (Γ, ψ), and γ ranges over Γ. The chapter first describes finitely many exceptional values, integration and the extension *K*(*x*), and approximating zeros of differential polynomials before proving Theorem 15.0.1, which states: If *K* is d-valued with ∂*K* = *K*, and *K* is a directed union of spherically complete grounded d-valued subfields, then *K* is newtonian. In concrete cases the hypothesis *K* = ∂*K* in the theorem can often be verified by means of Corollary 15.2.4.