Sala Seminari (Dip. Matematica).<\/p>\n
In this work we characterize the subsets of R^nthat areimages of Nash maps f : R^m \u2192 R^n. We prove Shiota\u2019s conjecture andshow that a subset S \u2282 R^nis the image of a Nash map f : R^m \u2192 R^nif and only if S is semialgebraic, pure dimensional of dimension d \u2264 mand there exists an analytic path \u03b1 : [0, 1] \u2192 S whose image meetsall the connected components of the set of regular points of S. Given asemialgebraic set S \u2282 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. <\/p>\n","protected":false},"excerpt":{"rendered":"
In this work we characterize the subsets of R^nthat areimages of Nash maps f : R^m \u2192 R^n. We prove Shiota\u2019s conjecture andshow that a subset S \u2282 R^nis the image of a Nash map f : R^m \u2192 R^nif and only if S is semialgebraic, pure dimensional of…<\/p>\n