Stratified-algebraic vector bundles

Kucharz, Wojciech; Kurdyka, Krzysztof

doi:10.1515/crelle-2015-0105

Cited by 28 publications

(66 citation statements)

References 47 publications

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“…It follows from Proposition 2.3 that stratified-regular functions coincide with continuous hereditarily rational functions studied in [44,45]. Furthermore, as explained in [29,61,66], stratified-regular maps defined on a constructible subset of a real algebraic variety are identical with regulous maps.…”

Section: Remark 25mentioning

confidence: 71%

“…By [61,Propositions 7.2 and 7.7], h is not homotopic to any stratified-regular map. In particular, h cannot be approximated by stratified-regular maps, which is of interest in view of Theorems 1.3 and 1.5.…”

Section: Example 17mentioning

confidence: 99%

“…In a somewhat less general context, S -algebraic and stratified-algebraic F-vector bundles are thoroughly investigated in [57,59,61,63,66]. Proposition 5.6 Let V be a real algebraic variety, X ⊆ V some subset, S a stratification of V , and ξ a topological F-vector subbundle of ε n X (F) for some n. Then the following conditions are equivalent:…”

Section: Lemma 53mentioning

confidence: 99%

“…2. Stratified-regular maps and functions are thoroughly investigated, in a more restrictive framework, in [4,29,30,44,45,50,52,54,55,58,[60][61][62]66,74,75,84], where they sometimes appear under different names (cf. Remark 2.5).…”

Section: Introductionmentioning

confidence: 99%

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Piecewise-regular maps

Kucharz

2017

Math. Ann.

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Let V , W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and X ⊆ V some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X . Furthermore, a continuous map f : X → W is said to be piecewise-regular if there exists a stratification S of V such that for every stratum S ∈ S the restriction of f to each connected component of X ∩ S is a regular map. By a stratification of V we mean a finite collection of pairwise disjoint Zariski locally closed subvarieties whose union is equal to V . Assuming that the subset X of V is compact, we prove that every continuous map from X into a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the variety V is compact and nonsingular, we prove that each continuous map from V into a unit sphere is homotopic to a piecewise-regular map of class C k , where k is an arbitrary nonnegative integer. Mathematics Subject Classification 14P05 · 14P99 · 57R22Communicated by Ngaiming Mok.

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Section: Remark 25mentioning

confidence: 71%

Section: Example 17mentioning

confidence: 99%

Section: Lemma 53mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Piecewise-regular maps

Kucharz

2017

Math. Ann.

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“…Both authors initiated a theory of vector bundles [9] on real algebraic varieties, in which continuous rational functions are used to define morphisms. Continuous rational functions, under the name fonctions régulues, are the object of investigation in [3].…”

Section: Introductionmentioning

confidence: 99%

Rationality of semialgebraic functions

Kucharz¹,

Kurdyka²

2017

Analytic and Algebraic Geometry 2

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Regulous vector bundles

Kucharz

Zieliński

2018

Mathematische Nachrichten

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Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified‐algebraic vector bundles. Furthermore, using new results on curve‐rational functions, we characterize regulous vector bundles among families of vector spaces parametrized by an affine regulous variety. We also study relationships between regulous and topological vector bundles.

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Stratified-algebraic vector bundles

Cited by 28 publications

References 47 publications

Piecewise-regular maps

Piecewise-regular maps

Rationality of semialgebraic functions

Regulous vector bundles

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