- Title Pages
- Epigraph
- Preface
- Conventions and Notations
- Leitfaden
- Dramatis Personæ
- Introduction and Overview
- Chapter One Some Commutative Algebra
- Chapter Two Valued Abelian Groups
- Chapter Three Valued Fields
- Chapter Four Differential Polynomials
- Chapter Five Linear Differential Polynomials
- Chapter Six Valued Differential Fields
- Chapter Seven Differential-Henselian Fields
- Chapter Eight Differential-Henselian Fields with Many Constants
- Chapter Nine Asymptotic Fields and Asymptotic Couples
- Chapter Ten <i>H</i>-Fields
- Chapter Eleven Eventual Quantities, Immediate Extensions, and Special Cuts
- Chapter Twelve Triangular Automorphisms
- Chapter Thirteen The Newton Polynomial
- Chapter Fourteen Newtonian Differential Fields
- Chapter Fifteen Newtonianity of Directed Unions
- Chapter Sixteen Quantifier Elimination
- Appendix A Transseries
- Appendix B Basic Model Theory
- Bibliography
- List of Symbols
- Index

# The Newton Polynomial

# The Newton Polynomial

- Chapter:
- (p.585) Chapter Thirteen The Newton Polynomial
- 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

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 equipped with a monomial group and (Γ, ψ) is the asymptotic couple of *K*. Throughout, *P* is an element of *K*{*Y*}superscript Not Equal To. The chapter first revisits the dominant part of *P* before discussing the elementary properties of the Newton polynomial. It then presents results about the shape of the Newton polynomial and considers realizations of three cuts in the value group Γ of *K*. It also describes eventual equalizers, along with further consequences of ω-freeness and λ-freeness, the asymptotic equation over *K*, and some special *H*-fields.

*Keywords:*
differential-valued field, dominant part, Newton polynomial, value group, eventual equalizer, ω-freeness, λ-freeness, asymptotic equation, H-field

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- Title Pages
- Epigraph
- Preface
- Conventions and Notations
- Leitfaden
- Dramatis Personæ
- Introduction and Overview
- Chapter One Some Commutative Algebra
- Chapter Two Valued Abelian Groups
- Chapter Three Valued Fields
- Chapter Four Differential Polynomials
- Chapter Five Linear Differential Polynomials
- Chapter Six Valued Differential Fields
- Chapter Seven Differential-Henselian Fields
- Chapter Eight Differential-Henselian Fields with Many Constants
- Chapter Nine Asymptotic Fields and Asymptotic Couples
- Chapter Ten <i>H</i>-Fields
- Chapter Eleven Eventual Quantities, Immediate Extensions, and Special Cuts
- Chapter Twelve Triangular Automorphisms
- Chapter Thirteen The Newton Polynomial
- Chapter Fourteen Newtonian Differential Fields
- Chapter Fifteen Newtonianity of Directed Unions
- Chapter Sixteen Quantifier Elimination
- Appendix A Transseries
- Appendix B Basic Model Theory
- Bibliography
- List of Symbols
- Index