Continuous rational maps into the unit 2-sphere

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.

Section: Introductionmentioning

confidence: 99%

Piecewise-regular maps

2017

Math. Ann.

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“…Continuous rational maps have only recently become the object of serious investigation, cf. [11,[17][18][19][20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Continuous rational maps into spheres

2016

Math. Z.

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“…This is justified since

double-struckF

‐vector bundles have already proved to be very useful in real algebraic geometry, cf. . In Section 5, we investigate which families of

double-struckF

‐vector spaces on an affine regulous (resp.…”

Section: Introductionmentioning

confidence: 99%

Regulous vector bundles

Mathematische Nachrichten

Zieliński

2018

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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.