# Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Patrick Bonckaert^{[1]}; Freek Verstringe^{[1]}

- [1] Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium)

Annales de l’institut Fourier (2012)

- Volume: 62, Issue: 6, page 2211-2225
- ISSN: 0373-0956

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topBonckaert, Patrick, and Verstringe, Freek. "Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part." Annales de l’institut Fourier 62.6 (2012): 2211-2225. <http://eudml.org/doc/251148>.

@article{Bonckaert2012,

abstract = {We explore the convergence/divergence of the normal form for a singularity of a vector field on $\mathbb\{C\}^n$ with nilpotent linear part. We show that a Gevrey-$\alpha $ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha $ type with the use of a Gevrey-$1+\alpha $ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.},

affiliation = {Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium); Universiteit Hasselt Agoralaan Gebouw D 3590 Diepenbeek (Belgium)},

author = {Bonckaert, Patrick, Verstringe, Freek},

journal = {Annales de l’institut Fourier},

keywords = {normal forms; nilpotent linear part; representation theory; Gevrey normalization},

language = {eng},

number = {6},

pages = {2211-2225},

publisher = {Association des Annales de l’institut Fourier},

title = {Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part},

url = {http://eudml.org/doc/251148},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Bonckaert, Patrick

AU - Verstringe, Freek

TI - Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

JO - Annales de l’institut Fourier

PY - 2012

PB - Association des Annales de l’institut Fourier

VL - 62

IS - 6

SP - 2211

EP - 2225

AB - We explore the convergence/divergence of the normal form for a singularity of a vector field on $\mathbb{C}^n$ with nilpotent linear part. We show that a Gevrey-$\alpha $ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha $ type with the use of a Gevrey-$1+\alpha $ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

LA - eng

KW - normal forms; nilpotent linear part; representation theory; Gevrey normalization

UR - http://eudml.org/doc/251148

ER -

## References

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