Quasianalytic Denjoy-Carleman classes and o-minimality

Rolin, Jean-Philippe; Speissegger, Patrick; Wilkie, A. J.

doi:10.1090/s0894-0347-03-00427-2

Cited by 120 publications

(68 citation statements)

References 20 publications

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“…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]. The latter surveys methods of constructing o-minimal structures and discusses the connections with "topologie modérée".…”

Section: O-minimal Structures Over Rmentioning

confidence: 99%

“…The latter surveys methods of constructing o-minimal structures and discusses the connections with "topologie modérée". In particular [80], there exist pairs of o-minimal structures that are incompatible in that their union is not contained in any o-minimal structure, and consequently there does not exist a "largest" o-minimal structure over R. Examples are given in [29] of natural functions that are not definable in R an,exp . For example the error function x exp(−t)dt/t on (0, ∞) are not definable in R an,exp , though their restrictions to any compact subinterval are, and they are definable in the o-minimal structure R Pfaff generated by Pfaffian functions (see e.g.…”

Section: O-minimal Structures Over Rmentioning

confidence: 99%

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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%

Section: O-minimal Structures Over Rmentioning

confidence: 99%

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

2011

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“…sets which are subanalytic or quasi-subanalytic in a semialgebraic compactification (see e.g. [11,7]). In turn, the locally definable sets are then precisely the subanalytic or quasisubanalytic ones.…”

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confidence: 99%

A theorem on generic intersections in an o-minimal structure

Nowak

2014

Fund. Math.

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“…Since o-minimal structures forbid the skullduggery used earlier to break rational numbers into their numerators and denominators, any o-minimal irrationality criterion should be an honest answer to Brun's question. Moreover, there are many o-minimal structures out there; indeed, there is no limit to them, in the sense of there being no largest structure of which every other o-minimal structure is a reduct (Rolin, Speissesser and Wilkie [11]). It is thus not without the bounds of reason that somewhere in this vast universe of well-behaved structures there exists a set and a function that can detect at least some irrational numbers, if not all of them.…”

Section: O-minimal Irrationality Criteriamentioning

confidence: 99%