Normal forms for formal series and germs of $C^\infty$-mappings with respect to the action of a group

This paper obtains a normal form for formal series and for germs of smooth mappings with respect to the action of a group. In particular, this yields a more precise version of the “resonance” normal form for differential equations. It is proved that under the action of a given group of $C^\infty$-mappings of coordinates any $C^\infty$-germ can be reduced to the sum of two germs, of which one is in normal form and the other has zero Taylor series at the origin.
Bibliography: 10 titles.