Linear equations on real algebraic surfaces

Kucharz, Wojciech; Kurdyka, Krzysztof

doi:10.1007/s00229-017-0925-8

Cited by 12 publications

(15 citation statements)

References 24 publications

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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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“…On the other hand, continuous rational functions on W = X(R) have only recently become the object of serious research. Their algebraic and geometric properties were considered in [12,13,15,24,29]. The homotopy and approximation properties of maps defined by continuous rational functions were studied in [16,17,18,19,21,31], and applications of such maps to algebraic and stratified-algebraic vector bundles were given in [4,20,22,23].Several examples discussed in [15] show that continuous rational functions on W = X(R) behave in a rather unusual way.…”

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confidence: 99%

Curve-rational functions

2017

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Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \, functions}$ (i.e., continuous rational on algebraic curves) and ${\it arc-rational\, functions}$ (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular, if $W$ is semialgebraic and $f$ is arc-rational, then $f$ is continuous and semialgebraic. We also show that an arc-rational function defined on an open set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study rational functions on products of varieties. As an application we obtain a characterization of regular functions. Finally, we get analogous results in the framework of complex algebraic varieties

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“…The case n = 1 is an easy exercise. The case n = 2 to which we turn next was settled by Kucharz and Kurdyka [10]. It is convenient to start with a result on Question 1 and the pointwise test.…”

Section: Continuous Semialgebraic and Regulous Solutionsmentioning

confidence: 99%