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In [[real algebraic geometry]], a '''Nash function''' on an open semialgebraic
subset ''U'' of '''R'''<sup>''n''</sup> is an [[analytic function]
''f'': ''U'' -> '''R''' satisfying a non trivial polynomial equation ''P''(''x'',''f''(''x'')) = 0
for all ''x'' in ''U''. (A
obtained from subsets of the form {''x'' in '''R'''<sup>''n''</sup> : ''P''(''x'')=0} or
{''x'' in '''R'''<sup>''n''</sup> : ''P''(''x'') > 0}, where ''P'' is a polynomial, by taking finite unions,
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order) is Nash on the open subset of symmetric matrices with no multiple
eigenvalue. Actually, Nash functions are those functions needed in order to have
an [[implicit function]] theorem in real algebraic geometry.
==Nash manifolds==
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semialgebraic analytic submanifolds of some '''R'''<sup>''n''</sup>. A Nash mapping
between Nash manifolds is then an analytic mapping with semialgebraic graph.
Nash functions and manifolds are named after [[John Forbes Nash|John F. Nash]], who proved (1952)
that any compact [[differentiable manifold|''C''<sup>∞</sup> manifold]] admits a Nash manifold structure,
i.e., is diffeomorphic to some Nash manifold. More generally, a
''C''<sup>∞</sup> manifold admits a Nash manifold strucure if and only if it
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==Local properties==
The local properties of Nash functions are well understood. The ring of [[germ (mathematics)|germs]] of
Nash functions at a point of a Nash manifold of dimension ''n'' is isomorphic to
the ring of algebraic power series in ''n'' variables (i.e., those series
satisfying a non trivial polynomial equation), which is the [[hensel's lemma|henselization]] of the
ring of germs of rational functions. In particular, it is a [[regular local ring]]
of dimension ''n''.
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The global properties are more difficult to obtain. The fact that the ring of
Nash functions on a Nash manifold (even non compact) is [[noetherian ring|noetherian]] was proved
independently (1973) by Jean-Jacques Risler and Gustave Efroymson. Nash
manifolds have properties similar to but weaker than [[Cartan's theorems A and B]]
on [[Stein manifold|Stein manifolds]]. Let <math>\mathcal{N}</math> denote the sheaf of Nash function germs on
a Nash manifold ''M'', and <math>\mathcal{I}</math> be a [[coherent sheaf]] of
<math>\mathcal{N}</math>-ideals. Assume <math>\mathcal{I}</math> is finite, i.e., there exists a
finite open semialgebraic covering <math>\{U_i\}</math> of ''M'' such that, for each ''i'',
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==Generalizations==
Nash functions and manifolds can be defined over any [[real closed field]] instead
of the field of real numbers, and the above statements still hold. Abstract Nash functions
can also be defined on the real spectrum of any commutative ring.
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