On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation

Popescu, Liliana Georgeta; Popescu, Nicolae

doi:10.21099/tkbjm/1496161567

Cited by 21 publications

(11 citation statements)

References 6 publications

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“…Then the residue field of w is the simple transcendental extension k v >(r*) of the residue field k v > of v (cf. [4] or [5]). As in [5,Cor.l.5], it can be easily verified that if F(x) e K [x] is such that w(F(x)) = 0, then F' e kAr'].…”

Section: A Note On Popescu's Resultsmentioning

confidence: 99%

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Generalized Hensel's lemma

Khanduja¹,

Saha

1999

Proceedings of the Edinburgh Mathematical Society

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Let (K, v) be a complete, rank-1 valued field with valuation ring Rv, and residue field kv. Let vx be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in Rv[x] are such that (i) vx(F(x) – G0(x)H0(x)) > 0, (ii) the leading coefficient of G0(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of vx satisfying vx(A(x)G0(x) + B(x)H0(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G0(x), (c) vx(G(x)–G0(x)) > 0, vx(H(x) – H0(x)) > 0. In this paper, our goal is to prove an analogous result when vx is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of kv.

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Section: A Note On Popescu's Resultsmentioning

confidence: 99%

“…Multiply both sides of (3) by P (x) and then substituting for B(x)P (x) and C(x)P (x) from (5) and 6, we obtain…”

Section: Proof Of Theorem 11 and Corollary 12mentioning

confidence: 99%

Generalized Hensel's lemma

Khanduja¹,

Saha

1999

Proceedings of the Edinburgh Mathematical Society

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“…By (9), for each j ∈ I we may take f j ∈ K[x] n such that H µ (f j ) = ζH µ (u) j ζ j . Then f = φ s (f 0 + · · · + f j φ je + · · · + f d φ de ) satisfies all our requirements.…”

Section: Residual Polynomial Operatormentioning

confidence: 99%

“…Some of the results of the paper can be found in [11], but only for augmented valuations. Also, in [9] some partial results are obtained for residually transcendental valuations, by using the fact that these valuations are determined by a minimal pair.…”

Section: Introductionmentioning

confidence: 99%

Key polynomials over valued fields

Nart¹

2020

Publicacions Matemàtiques

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Let K be a field. For any valuation µ on K[x] admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of µ-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of µ.2010 Mathematics Subject Classification: Primary: 13A18; Secondary: 13J10, 12J10.

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“…After MacLane's work, inductive valuations were rediscovered and extensively studied as residually transcendental extensions of v to K(x) [1,17,19]. In this approach, the valuations are first analyzed for algebraically closed fields, where they may be obtained as a simple augmentation of µ 0 with respect to a key polynomial of degree one.…”

Section: Introductionmentioning

confidence: 99%

Residual ideals of MacLane valuations

et al. 2015

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Abstract. Let K be a field equipped with a discrete valuation v. In a pioneering work, MacLane determined all valuations on K(x) extending v. His work was recently reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on K(x) and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.

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On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation

Abstract: We show that an algebraic immediate valuation ring extension of characteristic p > 0 is a filtered union of complete intersection algebras of finite type.

Cited by 21 publications

References 6 publications

Generalized Hensel's lemma

Generalized Hensel's lemma

Key polynomials over valued fields

Residual ideals of MacLane valuations

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