RT Journal Article
T1 Surjective Nash maps between semialgebraic sets
A1 Carbone, Antonio
A1 Fernando Galván, José Francisco
AB In this work we study the existence of surjective Nash maps between two given semialgebraic sets S and T. Some key ingredients are: the irreducible components S*ᵢ of S (and their intersections), the analytic path-connected components Tⱼ of T (and their intersections) and the relations between dimensions of the semialgebraic sets S*ᵢ and Tⱼ . A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps. The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: a full characterization of the semialgebraic subsets S ⊂ Rⁿ that are the image of the closed unit ball B ͫ of R ͫ centered at the origin under a Nash map f : R ͫ → Rⁿ. The necessary and sufficient conditions that must satisfy such a semialgebraic set S are: it is compact,connected by analytic paths and has dimension d ≤ m. Two remarkable consequences of the latter result are the following: (1) pure dimensional compact irreducible arcsymmetric semialgebraic sets of dimension d are Nash images of Bd, and (2) compact semialgebraic sets of dimension d are projections of non-singular algebraic sets of dimension d whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions).
PB Elsevier
SN 0001-8708
YR 2024
FD 2024
LK https://hdl.handle.net/20.500.14352/104459
UL https://hdl.handle.net/20.500.14352/104459
LA eng
NO Carbone, Antonio, and José F. Fernando. "Surjective Nash maps between semialgebraic sets." Advances in Mathematics 438 (2024): 109288.
NO 2023 Acuerdos transformativos CRUE
NO Ministerio de Ciencia e Innovación
NO Universidad Complutense de Madrid
DS Docta Complutense
RD 7 abr 2025