Nash function: Difference between revisions

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In [[real algebraic geometry]], a '''Nash function''' on an open semialgebraic subset ''U'' ⊂ '''R'''^''n'' is an [[analytic function]]

''f'': ''U'' → '''R''' satisfying a non trivial polynomial equation ''P''(''x'',''f''(''x'')) = 0 for all ''x'' in ''U'' (A [[semialgebraic set|semialgebraic subset]] of '''R'''^''n'' is a subset obtained from subsets of the form {''x'' in '''R'''^''n'' : ''P''(''x'')=0} or {''x'' in '''R'''^''n'' : ''P''(''x'') > 0}, where ''P'' is a polynomial, by taking finite unions, finite intersections and complements).

subset ''U'' of '''R'''^''n'' is an [[analytic function]]

''f'': ''U'' -> '''R''' satisfying a non trivial polynomial equation ''P''(''x'',''f''(''x'')) = 0

'''Examples'''

for all ''x'' in ''U''. (A [[semialgebraic set|semialgebraic subset]] of '''R'''^''n'' is a subset

finite intersections and complements.) *Polynomial and regular rational functions are Nash functions.

obtained from subsets of the form {''x'' in '''R'''^''n'' : ''P''(''x'')=0} or

function *<math>x\mapsto \sqrt{1+x^2}</math> is Nash on '''R'''; the function. which

{''x'' in '''R'''^''n'' : ''P''(''x'') > 0}, where ''P'' is a polynomial, by taking finite unions,

*the function which associates to a real symmetric matrix its ''i''-th eigenvalue (in increasing order) is Nash on the open subset of symmetric matrices with no multiple eigenvalue.

finite intersections and complements.) Polynomial and regular rational functions are

Nash functions; the

eigenvalue. Actually, Nash functions are those functions needed in order to have an [[implicit function]] theorem in real algebraic geometry.

function <math>x\mapsto \sqrt{1+x^2}</math> is Nash on '''R'''; the function which

associates to a real symmetric matrix its ''i''-th eigenvalue (in increasing

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==

Along with Nash functions one defines '''Nash manifolds''', which are semialgebraic analytic submanifolds of some '''R'''^''n''. A Nash mapping

between Nash manifolds is then an analytic mapping with semialgebraic graph. Nash functions and manifolds are named after [[John Forbes Nash, Jr.]], who proved (1952) that any compact [[differentiable manifold|''C''^&infin; manifold]] admits a Nash manifold structure, i.e., is diffeomorphic to some Nash manifold. More generally, a ''C''^∞ manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact ''C''^∞ manifold possibly with boundary. Nash's result was later (1973) completed by Alberto Tognoli who proved that any compact ''C''^∞ manifold is diffeomorphic to some affine real algebraic manifold ; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the ''C''^∞ and the algebraic categories.

semialgebraic analytic submanifolds of some '''R'''^''n''. A Nash mapping

between Nash manifolds is then an analytic mapping with semialgebraic graph.

Nash functions and manifolds are named after [[John Forbes Nash, Jr.]], who proved (1952)

that any compact [[differentiable manifold|''C''^&infin; manifold]] admits a Nash manifold structure,

i.e., is diffeomorphic to some Nash manifold. More generally, a

''C''^∞ manifold admits a Nash manifold structure if and only if it

is diffeomorphic to the interior of some compact ''C''^∞ manifold

possibly with boundary. Nash's result was later (1973) completed by Alberto

Tognoli who proved that any compact ''C''^∞ manifold is

diffeomorphic to some affine real algebraic manifold ; actually, any Nash

manifold is Nash diffeomorphic to an affine real algebraic manifold. These

results exemplify the fact that the Nash category is somewhat intermediate

between the ''C''^∞ and the algebraic categories.

==Local properties==