Logarithmic-exponential series

Dries, Lou van den; Macintyre, Angus; Marker, David

doi:10.1016/s0168-0072(01)00035-5

Cited by 61 publications

(129 citation statements)

References 19 publications

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“…Example 1.2. Let k be a logarithmic-exponential ordered field, and let F be a differential subfield of k t LE containing k x and closed under powers; that is, if 0 < f ∈ F and r ∈ k, then f r ∈ F. (See [5] for the construction of k t LE , the field of logarithmic-exponential series over k.) Let v be the valuation on F with valuation ring f ∈ F f ≤ r for some r ∈ k , and associate to F an H-couple just as we did for the Hardy fields above, with 1 = v t = v x −1 . (In Section 3 we show this gives indeed an Hcouple over k.)…”

Section: Definitions and Resultsmentioning

confidence: 99%

Closed asymptotic couples

Aschenbrenner

Dries

2000

Journal of Algebra

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The derivation of a Hardy field induces on its value group a certain function ψ. If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over . Such "ordered vector spaces with ψ-function" are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over . We analyze in detail finitely generated extensions of a given H-couple.

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Section: Definitions and Resultsmentioning

confidence: 99%

Closed asymptotic couples

Aschenbrenner

Dries

2000

Journal of Algebra

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“…This extension also naturally works for maximally complete fields and fields of LE-series, see also [DMM2] and [DMM3].…”

Section: Notation and Quantifier Eliminationmentioning

confidence: 95%

Real closed fields with non-standard and standard analytic structure

Cluckers¹,

Lipshitz²,

Robinson³

2008

Journal of the London Mathematical Society

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Abstract. We consider the ordered field which is the completion of the Puiseux series field over R equipped with a ring of analytic functions on [−1, 1] n which contains the standard subanalytic functions as well as functions given by tadically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields Rn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [HP] of a sentence which is not true in any o-minimal expansion of R (shown in [LR3] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in Rn, but not true in any o-minimal expansion of any of the fields R, R 1 , . . . , R n−1 .

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“…Other sources for the definitions are: [1], [5], [8], [15]. I will generally follow the notation from [9].…”

Section: Reviewmentioning

confidence: 99%

“…I will generally follow the notation from [9]. The well-based version of the construction is described in [8] (or [10, Def. 2.1]).…”

Section: Reviewmentioning

confidence: 99%

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Fractional iteration of series and transseries

Edgar

2013

Trans. Amer. Math. Soc.

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We investigate compositional iteration of fractional order for transseries. For any large positive transseries T of exponentiality 0, there is a family T [s] indexed by real numbers s corresponding to iteration of order s. It is based on Abel's Equation. We also investigate the question of whether there is a family T [s] all sharing a single support set. A subset of the transseries of exponentiality 0 is divided into three classes ("shallow", "moderate" and "deep") with different properties related to fractional iteration.

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Logarithmic-exponential series

Cited by 61 publications

References 19 publications

Closed asymptotic couples

Closed asymptotic couples

Real closed fields with non-standard and standard analytic structure

Fractional iteration of series and transseries

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