RT Journal Article
T1 Uniform bounds on complexity and transfer of global properties of Nash functions
A1 Coste, M.
A1 Ruiz Sancho, Jesús María
A1 Shiota, Masahiro
AB We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian.
PB Walter de Gruyter & co
SN 0075-4102
YR 2001
FD 2001-07
LK https://hdl.handle.net/20.500.14352/57889
UL https://hdl.handle.net/20.500.14352/57889
LA eng
NO DGICYT
DS Docta Complutense
RD 7 abr 2025