Hokkaido Mathematical Journal

$C^{\ell}$-$G$-triviality of map germs and Newton polyhedra, $G = \mathcal R$, $\mathcal C$ and $\mathcal K$

Marcelo José SAIA and Carlos Humberto Soares JÚNIOR

Full-text: Open access

Abstract

We provide estimates for the $C^{\ell}$-$G$-triviality, for $ 0 \leq \ell < \infty$ and $G$ is one of Mather's groups ${\mathcal R}$, ${\mathcal C}$ or ${\mathcal K}$, of deformations of analytic map germs $f: (\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ of type $f_t(x)=f(x)+θ(x,t)$ which satisfy a non-degeneracy condition with respect to some Newton polyhedron. We apply the method of construction of controlled vector fields and, for each group $G$, the control function is determined from the choice of a convenient {\it Newton filtration } in the ring of real analytic germs. The results are given in terms of the filtration of the coordinate function germs $f_1, \ldots , f_p$ of $f$.

Article information

Source
Hokkaido Math. J., Volume 37, Number 2 (2008), 331-348.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1253539558

Digital Object Identifier
doi:10.14492/hokmj/1253539558

Mathematical Reviews number (MathSciNet)
MR2415904

Zentralblatt MATH identifier
1207.58032

Subjects
Primary: 58C27

Keywords
$C^{\ell}$-determinacy Newton filtration controlled vector fields

Citation

SAIA, Marcelo José; JÚNIOR, Carlos Humberto Soares. $C^{\ell}$-$G$-triviality of map germs and Newton polyhedra, $G = \mathcal R$, $\mathcal C$ and $\mathcal K$. Hokkaido Math. J. 37 (2008), no. 2, 331--348. doi:10.14492/hokmj/1253539558. https://projecteuclid.org/euclid.hokmj/1253539558


Export citation