Rational maps in real algebraic geometry

Kucharz, Wojciech

doi:10.1515/advgeom.2009.024

Cited by 51 publications

(91 citation statements)

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“…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%

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: Introductionmentioning

confidence: 99%

Piecewise-regular maps

Kucharz

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%

“…Actually, the sets R(X, S 1 ) and R 0 (X, S 1 ) have equal closures in C(X, S 1 ), cf. [18]. Furthermore, the set [19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Continuous rational maps into spheres

Kucharz

2016

Math. Z.

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“…Continuous rational maps form a natural intermediate class between regular and continuous semi-algebraic maps, with many specific properties, cf. [3,[5][6][7].…”

Section: Introductionmentioning

confidence: 99%