Un théorème de quasi-transversalité. (A quasi-transversality theorem). (French) Zbl 0605.58005

Following the author’s remark, the essence of the problem can be outlined as follows: A smooth mapping \(f: X\to Y\) of an open subset \(X\subset {\mathbb{R}}^ n\) into an open subset \(Y\subset {\mathbb{R}}^ p\) is called quasi-inverse to a submanifold Z of Y if f transverse to Z (in the common sense of word) at all points with exception of an isolated (discrete) subset D of X. Let moreover \(Z_{\gamma}\) (\(\gamma\in \Gamma)\) be a family of submanifolds of Y satisfying certain Noetherian type conditions. Then the family of mappings f that are quasi-transverse to every \(Z_{\gamma}\) is dense in the space \(C^{\infty}(X,Y)\) of all smooth mappings.
Complete realization of these ideas is made in the space of jets of infinite order and formal power series over \({\mathbb{R}}\) and \({\mathbb{C}}\) using a large amount of new concepts from the commutative algebra on projective limit spaces (constructive sets, stratifications, analytical Noetherian algebras) which are of independent interests. We can mention only the main concept here: Let \(I\subset k[[ x_ i,y^{\omega}_ j]]\) \((i=1,...,n\); \(j=1,...,p\); \(| \omega | \leq q\); \(y_ j^{\omega}\) are the jet coordinates) be an ideal, \(f\in k[[ x_ i]]^ p\) a formal mapping. Denote \(\omega^ f_ k({\mathcal I})=\sqrt{J^ f_ k({\mathcal I})/J_ k({\mathcal I})}\), where \(J^ f_ k({\mathcal I})\), \(J_ k({\mathcal I})\) are certain ideals generated by \({\mathcal I}\) and certain Jacobians. If \(ht(J^{q+1}f)^*(\omega^ f_ k({\mathcal I}))\geq n\), then the jet \(J^ qf\) is called k-quasi-transverse to \({\mathcal I}\). If this is true for every field k, we speak of a quasi-transversality.
Reviewer: J.Chrastina


58A20 Jets in global analysis
57R99 Differential topology