Quantifier elimination in quasianalytic structures via non-standard analysis

Nowak, Krzysztof

doi:10.4064/ap114-3-4

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…In general, however, it is not true that assumption (b) (i.e. the quasianalyticity axiom (3)) for implies (b) for [Now15, Example 6.6].…”

Section: Quasianalytic Classesmentioning

confidence: 99%

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Composite quasianalytic functions

Silva¹,

Bierstone²,

Chow³

2018

Compositio Math.

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We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the $o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

show abstract

“…In general, however, it is not true that assumption (b) (i.e. the quasianalyticity axiom (3)) for implies (b) for [Now15, Example 6.6].…”

Section: Quasianalytic Classesmentioning

confidence: 99%

“…In particular, in general, . Moreover, is the smallest Denjoy–Carleman class containing all such that [Now15, Remark 6.2].…”

Section: Quasianalytic Classesmentioning

confidence: 99%

Composite quasianalytic functions

Silva¹,

Bierstone²,

Chow³

2018

Compositio Math.

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“…the expansion of the field R by restricted analytic functions) or, more generally, a quasianalytic structure (i.e. the expansion of the field R by restricted quasianalytic functions; see e.g [9,5,7]…”

mentioning

confidence: 99%

On Intersections of Generic Perturbations of Definable Sets

Mycielski

Nowak

2016

Bull. Polish Acad. Sci. Math.

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Consider an o-minimal expansion R of a real closed field R and two definable sets E and M. We introduce concepts of a locally transitive (abbreviated to l.t.) and a strongly locally transitive (abbreviated to s.l.t.) action of E on M. In the former case, M is supposed to be of pure dimension m; in the latter, both M and E are supposed to be of pure dimension. We treat the elements of E as perturbations of the set M. We prove that if E acts l.t. on M , and A and B are two non-empty definable subsets of M of dimension dim A ≤ dim B < dim M , then dim(σ(A) ∩ B) < dim A for a generic σ in E; here dim ∅ = −1. And if E acts s.l.t. on M and A and B are two definable subsets of M , then dim(σ(A) ∩ B) ≤ max{dim A + dim B − m, −1} for a generic σ in E. We give an example of a l.t. action E on M for which the latter conclusion of the intersection theorem fails. We also prove a theorem on the intersections of generic perturbations in terms of the exceptional set T ⊂ M of points at which E is not l.t. Finally, we provide some natural conditions which imply that T is a nowhere dense subset of M .

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Solutions of quasianalytic equations

Silva

Biborski

Bierstone

2017

Sel. Math. New Ser.

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Abstract. The article develops techniques for solving equations G(x, y) = 0, where G(x, y) = G(x 1 , . . . , xn, y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of C ∞ functions definable in a polynomially-bounded o-minimal structure). We show that, if G(x, y) = 0 has a formal power series solution y = H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y = h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

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Quantifier elimination in quasianalytic structures via non-standard analysis

Cited by 5 publications

References 8 publications

Composite quasianalytic functions

Composite quasianalytic functions

On Intersections of Generic Perturbations of Definable Sets

Solutions of quasianalytic equations

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