Regulous vector bundles

Abstract

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.

Piecewise-regular maps

“…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.…”

“…[6,39]. This is also the case for stratified-algebraic F-vector bundles introduced in [61] and further investigated in [57,59,63,66]. Theorems 1.3 and 1.4 have a bearing on F-vector bundles as well, which is elaborated upon in Sect.…”

“…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:…”

“…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).…”

Nash regulous functions

“…Such functions, which often appear under different names and in more general contexts, have several remarkable properties and applications, cf. [2,5,6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]27,29]. In particular, the authors of [5] obtained the following: a variant of the classical Nullstellensatz for the ring R k (R n ) of k-regulous functions on R n , a description of the zero locus Z(F ) of an arbitrary collection F ⊆ R k (R n ) in terms of Zariski (algebraically) constructible sets, and counterparts of Cartan's theorems A and B for quasi-coherent k-regulous sheaves.…”