p-adic and Real Subanalytic Sets

Denef, Jan; Dries, Lou van den

doi:10.2307/1971463

Cited by 212 publications

(233 citation statements)

References 14 publications

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“…From now on in Section 3, definable sets and functions will be so for the language L. Note that the study of definable sets was initiated in the works of Macintyre [25] and Denef and van den Dries [19] in the p-adic case, and was generalized later to this and other settings in for example [1,12,14,36].…”

Section: Non-archimedean Yomdin-gromov Parametrizations With Taylor Amentioning

confidence: 99%

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

2015

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We prove an analog of the Yomdin-Gromov lemma for p-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of p-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over C((t)), in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

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Section: Non-archimedean Yomdin-gromov Parametrizations With Taylor Amentioning

confidence: 99%

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

2015

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“…The result we need can be found in [13] where it is formulated in terms of so-called "finitely sub-analytic sets". Since these are exactly the sets definable in R an (see [3]) we may reformulate the result as follows…”

Section: Corollarymentioning

confidence: 99%

“…The fact that we do not obtain non-sub-analytic sets is due, of course, to our original truncation of the analytic functions.) In this form Gabrielov's theorem was given a fairly straightforward treatment, based on the Weierstrass preparation theorem and Tarski's elimination theory, by Denef and van den Dries ( [3]). Thus, although we do not have full quantifier elimination for this local analytic structure (together with the ordered ring structure) on R, we do have elimination down to existential formulas.…”

Section: Introductionmentioning

confidence: 99%

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

Wilkie¹

1996

J. Amer. Math. Soc.

323

153

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“…The construction was given in our preprints [25]. This indicates that the classical theorem by J. Denef and L. van den Dries [6] does no longer hold for quasianalytic structures. More precisely, we construct a plane definable curve, which also shows that the classical theorem of Lojasiewicz [14] that every subanalytic curve is semianalytic is no longer true for quasianalytic structures.…”

Section: Introductionmentioning

confidence: 99%

Quantifier elimination in quasianalytic structures via non-standard analysis

Nowak

2015

Ann. Polon. Math.

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Abstract. The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a quasianalytic Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function 1/x. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as Lojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces.

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p-adic and Real Subanalytic Sets

Cited by 212 publications

References 14 publications

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

Quantifier elimination in quasianalytic structures via non-standard analysis

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