The Elementary Theory of Restricted Analytic Fields with Exponentiation

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

doi:10.2307/2118545

Cited by 231 publications

(282 citation statements)

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“…[30]) the set Γ exp is not subanalytic at infinity, so is not contained in R an , while R an contains the graphs of restricted analytic functions such as {(x, y) ∈ R 2 : y = sin(x), x ∈ [−1, 1]} that are not definable in R exp (see [12]). However the structure R an,exp generated by the union of R an and R exp is o-minimal (van den Dries and Miller [31]; see also [28]). Further examples may be found described in [80], [86], [79].…”

Section: O-minimal Structures Over Rmentioning

confidence: 99%

O-minimality and the André-Oort conjecture for C^n

2011

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We give an unconditional proof of the André-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the ManinMumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink's relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of André-OortManin-Mumford type by Pila.

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Section: O-minimal Structures Over Rmentioning

confidence: 99%

O-minimality and the André-Oort conjecture for C^n

2011

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“…Besides being a universal domain for ordered fields (in the sense that every ordered field whose domain is a set can be embedded in No), it admits an exponential function exp : No → No [Gon86] and an interpretation of the real analytic functions restricted to finite numbers [All87], making it, thanks to the results of [Res93,DMM94], into a model of the theory of the field of real numbers endowed with the exponential function and all the real analytic functions restricted to a compact box [DE01].…”

Section: Introductionmentioning

confidence: 99%

Surreal numbers, derivations and transseries

Berarducci

Mantova

2018

J. Eur. Math. Soc.

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Abstract. Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.

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“…The o-minimality of this structure was first established by van den Dries and Miller [46] and then the structure of its definable sets was more thoroughly explored by van den Dries, Macintyre, and Marker [45]. It is this structure which is relevant to the diophantine applications mentioned in the abstract.…”

Section: Thomas Scanlonmentioning

confidence: 98%