Nash function: Difference between revisions

Revision as of 20:36, 16 March 2011

Revision as of 17:54, 21 August 2012

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 non trivialnontrivial 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:

*Polynomial and regular rational functions are Nash functions.

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

Line 13:

==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 trivialnontrivial 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 non compactnoncompact) 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

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>