On a proof to Shiota’s conjecture to characterize Nash images of Euclidean spaces – José Fernando (Universidad Complutense de Madrid)

Venue

Sala Seminari (Dip. Matematica).

Abstract

In this work we characterize the subsets of R^nthat areimages of Nash maps f : R^m → R^n. We prove Shiota’s conjecture andshow that a subset S ⊂ R^nis the image of a Nash map f : R^m → R^nif and only if S is semialgebraic, pure dimensional of dimension d ≤ mand there exists an analytic path α : [0, 1] → S whose image meetsall the connected components of the set of regular points of S. Given asemialgebraic set S ⊂ R^nsatisfying the previous properties, we provide atheoretical strategy to construct (after Nash approximation) a Nash mapwhose image is the semialgebraic set S. This strategy includes resolutionof singularities, relative Nash approximation on Nash manifolds withboundary and other tools (such as the drilling blow-up) constructedad hoc for Nash manifolds and Nash subsets that may have furtherapplications to approach new problems. Some remarkable consequences are the following: (1) pure dimen- sional irreducible semialgebraic sets of dimension d with arc-symmetricclosure are Nash images of R^d; and (2) semialgebraic sets are projectionsof irreducible algebraic sets whose connected components are Nash diffeomorphic to Euclidean spaces.

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