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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 nontrivial polynomial equation ''P''(''x'',''f''(''x'')) = 0 for all ''x'' in ''U'' (A semialgebraic subset of R''n'' is a subset obtained from subsets of the form or , where ''P'' is a polynomial, by taking finite unions, finite intersections and complements). Some examples of Nash functions: *Polynomial and regular rational functions are Nash functions. * 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. 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 smooth manifold admits a Nash manifold structure, i.e., is diffeomorphic to some Nash manifold. More generally, a smooth manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact smooth manifold possibly with boundary. Nash's result was later (1973) completed by Alberto Tognoli who proved that any compact smooth 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 smooth and the algebraic categories. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nash functions」の詳細全文を読む スポンサード リンク
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