Big denominators and analytic normal forms: With an appendix by M. Zhitomirskii

Laurent Stolovitch

doi:10.1515/crelle-2013-0111

Article

Big denominators and analytic normal forms

With an appendix by M. Zhitomirskii

Laurent Stolovitch

Published/Copyright: January 10, 2014

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2016 Issue 710

Abstract

We study the regular action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object F₀ and we are interested in an analytic normal form for the whole affine space {F0+h.o.t.}. We prove that if the cohomological operator defined by F₀ has the big denominators property and if a formal normal form is well chosen, then this formal normal form holds in analytic category. We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. Moreover, we prove that if the denominators grow “relatively fast”, but not fast enough to satisfy the big denominators property, then we have a normal form, respectively local solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by explanation of known results and by new results in the problems of local classification of singularities of vector fields, non-isolated singularities of functions, tuples of germs of vector fields, local Riemannian metrics and conformal structures.

Funding source: ANR

Award Identifier / Grant number: ANR-10-BLAN 0102

Funding source: Israel Science Foundation

Award Identifier / Grant number: 1383/07

Received: 2013-7-5

Revised: 2013-10-21

Published Online: 2014-1-10

Published in Print: 2016-1-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2013-0111