-parametrization in O-minimal Structures

Kocel-Cynk, Beata; Pawłucki, Wiesław; Valette, Anna

doi:10.4153/cmb-2018-030-x

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…In both Pila-Wilkie's and Burguet's papers, the problem is resolved by an additional approximation argument on C r -smooth maps. We also remark that Kocel-Cynk, Pawlucki and Vallete have given a proof based on a somewhat different approach in the general o-minimal setting [5].…”

Section: The Yomdin-gromov Lemmamentioning

confidence: 99%

The Yomdin-Gromov algebraic lemma revisited

Binyamini

Novikov

2020

Preprint

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In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub's entropy conjecture for C ∞ maps. The statement was further refined by Gromov, producing what is now known as the Yomdin-Gromov algebraic lemma. Several complete proofs based on Gromov's sketch have appeared in the literature, but these have been considerably more complicated than Gromov's original presentation due to some technical issues.In this note we give a proof that closely follows Gromov's original presentation. We prove a somewhat stronger statement, where the parameterizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.

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Section: The Yomdin-gromov Lemmamentioning

confidence: 99%

The Yomdin-Gromov algebraic lemma revisited

Binyamini

Novikov

2020

Preprint

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Strict $${\mathcal {C}}^p$$-triangulations of sets locally definable in o-minimal structures with an application to a $$\mathcal C^p$$-approximation problem

Pawłucki

2023

Rev Mat Complut

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We show how to derive triangulations of sets locally definable in o-minimal structures from triangulations of compact definable sets. We give it in particular for strict $$\mathcal C^p$$ C p -triangulations which has been recently studied by the author. This combined with a theorem of Fernando and Ghiloni implies that every continuous mapping defined on a locally compact subset B of $$\mathbb R^m$$ R m with values in any locally definable and locally compact subset A of $$\mathbb R^n$$ R n can be approximated by $$\mathcal C^p$$ C p -mappings defined on B with values in A for any positive integer p.

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