Finitely smooth polynomial normal forms of \(C^{\infty}\)-diffeomorphisms in the neighborhood of a fixed point (Q749740)

Let \(f: X\to X\) be a \(C^{\infty}\)-map of a finite dimensional real space \(X\) into itself such that \(f(0)=0\). Consider the so-called polynomial resonant \(C^ k\)-normal form of the map \(f\) in the neighbourhood of the origin, i.e. the transformation \(f_ 1(x)=Lx+p_ 1(x),\) where \(L\) is a certain hyperbolic linear operator and \(p_ 1(x)\) is a certain polynomial [\textit{P. F. Hartman}, Ordinary differential equations. New York: Wiley (1964; Zbl 0125.32102)]. In this note a further simplification of this normal form is announced -- the details are too technical to state here. Analogous results for vector fields were obtained earlier by \textit{V. S. Samovol} [Tr. Mosk. Mat. O.-va 38, 187--219 (1979; Zbl 0438.34032) and Differ. Uravn. 24, No. 12, 2180--2184 (1988)].