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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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The Newton Polynomial

The Newton Polynomial

(p.585) Chapter Thirteen The Newton Polynomial
Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

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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