On a proof to shiota's conjecture to characterize nash images of Euclidean spaces

Luogo:  Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"
Ore 16:00

• Relatore:  Fernando Galvan Josè Francisco (Departamento de Algebra, Facultad de Ciencias Matematicas, Universidad  ́Complutense de Madrid)

Abstract:
In this work we characterize the subsets of R n that are images of Nash maps f : R m → R n . We prove Shiota’s conjecture and show that a subset S ⊂ R n is the image of a Nash map f : R m → R n if and only if S is semialgebraic, pure dimensional of dimension d ≤ m and there exists an analytic path α : [0, 1] → S whose image meets all the connected components of the set of regular points of S. Given a semialgebraic set S ⊂ R n satisfying the previous properties, we provide a theoretical strategy to construct (after Nash approximation) a Nash map whose image is the semialgebraic set S. This strategy includes resolution of singularities, relative Nash approximation on Nash manifolds with boundary and other tools (such as the drilling blow-up) constructed ad hoc for Nash manifolds and Nash subsets that may have further applications to approach new problems. Some remarkable consequences are the following: (1) pure dimensional irreducible semialgebraic sets of dimension d with arc-symmetric closure are Nash images of R d ; (2) semialgebraic sets are projections of irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces; and (3) compact d-dimensional smooth manifolds with boundary are smooth images of R d 

Referente: Riccardo Ghiloni