
Contents
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13.1 REVISITING THE DOMINANT PART 13.1 REVISITING THE DOMINANT PART
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Elementary facts on the dominant part Elementary facts on the dominant part
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Lemmas on Newton polynomials Lemmas on Newton polynomials
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Behavior under additive and multiplicative conjugation Behavior under additive and multiplicative conjugation
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13.3 THE SHAPE OF THE NEWTON POLYNOMIAL 13.3 THE SHAPE OF THE NEWTON POLYNOMIAL
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Statement of results Statement of results
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The l-free case The l-free case
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The w-free case The w-free case
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Newton polynomials and upward shift Newton polynomials and upward shift
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13.4 REALIZING CUTS IN THE VALUE GROUP 13.4 REALIZING CUTS IN THE VALUE GROUP
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The cut –< The cut –<
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PROOF. The first identity follows from Corollary 13.2.12. As this holds for every P, we get –Khyi = Г + Zvy + Z L(vy). Set α := vy, β := L(vy), and –1 = Г + Zβ. Since K has rational asymptotic integration by Corollary 11.6.8, and < β < (–>)0, we get from Corollary 9.8.6 that kβ 2 =Г for all nonzero k 2 Z, and –. [–1] = [–], so L–6=1 = . With 1 the restriction of L to –6=we get an H-1 asymptotic –. couple (–1; 1) with gap β. Since 0 < njβj < –>1for all {n > 1, and Ljβj= β, this gives the desired result in view of [–1 + Zβ] = [–] [ [β]. ∈Λ PROOF. The first identity follows from Corollary 13.2.12. As this holds for every P, we get –Khyi = Г + Zvy + Z L(vy). Set α := vy, β := L(vy), and –1 = Г + Zβ. Since K has rational asymptotic integration by Corollary 11.6.8, and < β < (–>)0, we get from Corollary 9.8.6 that kβ 2 =Г for all nonzero k 2 Z, and –. [–1] = [–], so L–6=1 = . With 1 the restriction of L to –6=we get an H-1 asymptotic –. couple (–1; 1) with gap β. Since 0 < njβj < –>1for all {n > 1, and Ljβj= β, this gives the desired result in view of [–1 + Zβ] = [–] [ [β]. ∈Λ
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The cut –6 The cut –6
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The residue field of Khyi The residue field of Khyi
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Combining multiplicative and compositional conjugation Combining multiplicative and compositional conjugation
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Notes and comments Notes and comments
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13.5 EVENTUAL EQUALIZERS 13.5 EVENTUAL EQUALIZERS
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More general multiplicative-compositional conjugations More general multiplicative-compositional conjugations
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Transition from ndeg to nmul Transition from ndeg to nmul
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The Newton diagram of P The Newton diagram of P
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Let (i0; : : : ; in) be as in Proposition 13.5.7. This tuple is uniquely determined by the data K, P, E. If mul P = ndegE P, then n = 0 and this tuple is just (mul P). To simplify notation, set em := e(P; im–1; im) for 1 6 m 6 n. We now have a complete description of the behavior of nmul P×g and ndeg P×g for g 2 E: Let (i0; : : : ; in) be as in Proposition 13.5.7. This tuple is uniquely determined by the data K, P, E. If mul P = ndegE P, then n = 0 and this tuple is just (mul P). To simplify notation, set em := e(P; im–1; im) for 1 6 m 6 n. We now have a complete description of the behavior of nmul P×g and ndeg P×g for g 2 E:
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Notes and comments Notes and comments
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13.6 FURTHER CONSEQUENCES OF w-FREENESS 13.6 FURTHER CONSEQUENCES OF w-FREENESS
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Notes and comments Notes and comments
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Cases of low complexity Cases of low complexity
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Notes and comments Notes and comments
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13.8 Asymptotic Equations 13.8 Asymptotic Equations
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Behavior of unravelers under immediate extensions Behavior of unravelers under immediate extensions
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13.9 SOME SPECIAL H-FIELDS 13.9 SOME SPECIAL H-FIELDS
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The ambient H-field L The ambient H-field L
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LEMMA 13.9.2. Let G 6= f0g be an ordered subgroup of v(M) such that (G6=) ⊆ G and G< is coinitial in v(M)<. Then G = v(M). LEMMA 13.9.2. Let G 6= f0g be an ordered subgroup of v(M) such that (G6=) ⊆ G and G< is coinitial in v(M)<. Then G = v(M).
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which suggests vg = –(e0+e1+e2+… ). which suggests vg = –(e0+e1+e2+… ).
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Realizing Rhgi as a Hardy field Realizing Rhgi as a Hardy field
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Notes and comments Notes and comments
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Thirteen The Newton Polynomial
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Published:June 2017
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Abstract
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.
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