Nash function: Difference between revisions

Revision as of 18:28, 13 March 2008

Latest revision as of 18:49, 23 December 2024

(10 intermediate revisions by 8 users not shown)

Line 1:

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 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). Some examples of Nash functions:

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

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

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

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

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

*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.

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,

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

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

Nash functions; the

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|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.

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|John F. Nash]], 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==

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 nontrivial 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''.

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''.

==Global properties==

The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold (even noncompact) 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]]s. Let <math>\mathcal{N}</math> denote the sheaf of Nash function germs on

The global properties are more difficult to obtain. The fact that the ring of

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'', <math>\mathcal{I}|_{U_i}</math> is generated by Nash functions on <math>U_i</math>. Then <math>\mathcal{I}</math> is globally generated by Nash functions on ''M'', and the natural map

Nash functions on a Nash manifold (even non compact) is [[noetherian ring|noetherian]] was proved

:::<math>H^0(M,\mathcal{N}) \to H^0(M,\mathcal{N}/\mathcal{I})</math>

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]]s. 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'',

<math>\mathcal{I}|_{U_i}</math> is generated by Nash functions on <math>U_i</math>. Then <math>\mathcal{I}</math>

is globally generated by Nash functions on ''M'', and the natural map

:<math>H^0(M,\mathcal{N}) \to H^0(M,\mathcal{N}/\mathcal{I})</math>

is surjective. However

:::<math>H^1(M,\mathcal{N})\neq 0</math>, \ \text{if} \ \dim ''(M'') > 0, </math>
contrarily to the case of Stein manifolds.

manifolds.

==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.

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.

== Sources ==

#J. Bochnak, M. Coste and M-F. Roy: Real algebraic geometry. Springer, 1998.

#M. Coste, J.M. Ruiz and M. Shiota: Global problems on Nash functions. Revista Matem\'aticaMatemática Complutense 17 (2004), 83--115.

#G. Efroymson: A Nullstellensatz for Nash rings. Pacific J. Math. 54 (1974), 101--112.

#J.F. Nash : Real algebraic manifolds. Annals of Mathematics 56 (1952), 405--421.

Line 74 ⟶ 38:

#A. Tognoli: Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27 (1973), 167--185.

[[Category:Real algebraic geometry]]