0 \}, \\ \\ & \Lambda =\max\,\{ -\theta_i : \theta_i<0\},\quad \lambda =\min\,\{ -\theta_i : \theta_i <0 \}. \end{array} $$ \medskip {\bf Lemma 8.1.}\quad {\it Let $\xi$ preserve the standard volume form and $F$ be its hamiltonian defined by} (2.1). {\it Let $k$ be an integer satisfying the inequality $$ K>\frac{2\Lambda}{\lambda}+\frac{2\hbox{\rm M}}{\mu}+(k+1) (\frac{\hbox{\rm M}}{\lambda}+\frac{\Lambda}{\mu})+2. \eqno {\rm (8.1)} $$ Then there exists a canonical transformation of class $C^k$ defined in the vicinity of the origin which conjugates $F$ and $j_0^K(F)$.} \medskip {\bf Proof.} Identify the jet $j_0^{K-1}(\xi)$ with the corresponding Taylor's polynomial. Clearly, the polynomial vector field $j_0^{K-1}(\xi)$ respects the standard volume form. Recall that the hamiltonian $F$ is a $(d-2)$-form of class $C^K$. It easily follows that $j_0^K(F)$ is the hamiltonian for $j_0^{K-1}(\xi)$. Consider the cohomology equation (5.1) and the related affine extension (5.2), where $\xi_{\varepsilon}= \varepsilon j_0^{K-1}(\xi)+(1-\varepsilon )\xi,\ G=j_0^K(F)$. Then $\A=(\A_{pq}(u))$ is a family of linear morphisms of class $C^{K-1}\subset C^{k+1}$ and $j_0^K(G-F)=0$. It is easily seen from (5.1) that the spectrum of the matrix $\A(0)$ lies between the lines $\hbox{\rm Re}\nu =-2\Lambda$ and $\hbox{\rm Re}\nu =2\hbox{\rm M}$. Now apply Theorem 2${}^\prime$ from [4]. The proof is complete. \medskip {\bf Lemma 8.2.}\quad {\it Let $\xi$ be a symplectic vector field on $\R^{2d}$, $F$ be its hamiltonian, $k$ be an integer satisfying $$ K>\frac{2\Lambda (k+1)}{\lambda}+2. \eqno {\rm (8.2)} $$ Then there exists a (locally defined) canonical change of variables of class $C^k$ which conjugates $F$ and $j_0^K(F)$.} \medskip The proof is similar to that of Lemma 8.1. Observe that the spectrum of the linear part of $\xi$ is symmetric relatively the imaginary axis and the first eigenvalue of (5.4) equals to 0. Now let $\xi$ be a contact vector field on $\R^{2d+1}$, $F$ be its contact hamiltonian and $\eta$ be the symplectic vector field on $\R^{2d+2}$ that corresponds to $\xi$. Since $\xi (0)=0$, we have $\eta (0)=0$ and the origin is a hyperbolic singular point of $\eta$. Denote $A=D\eta (0)$. Define the numbers $\lambda$ and $\Lambda$ for $A$, as before. \medskip {\bf Lemma 8.3.}\quad {\it If $k$ satisfies the inequality} (8.2) {\it then there exists a canonical transformation of class $C^k$ which conjugates $F$ and $j_0^K(F)$ near the origin.} \medskip Combining these results with the results presented in the preceding section, we obtain \medskip {\bf Theorem 8.1.}\quad {\it Let $\xi$ be a vector field of class $C^K$ which preserves the (standard) volume form (symplectic form, contact structure) and the origin be a hyperbolic singular point of $\xi$. Let $F$ be the corresponding local hamiltonian. If $k$ satisfies the inequality} (8.1) {\it or, respectively}, (8.2), {\it then there exists a local canonical transformation of class $C^k$ which reduces the hamiltonian $F$ (and, consequently, the vector field $\xi$) to the polynomial resonant normal form. } \section{Simplified resonant normal forms} It is known (see, for example, [6,2]) that some resonant monomials $x^{\tau}$ can be eliminated by using $C^k$ smooth changes of variables. Let us present the following sufficient condition that comprises all the known ones (see [2]). \medskip {\bf Condition $\a (k)$.} The multi-index $\tau =(\tau^1,\ldots,\tau^n ) \in \Z^n_{+}$ is said to satisfy the condition $\a (k),\ k\ge 1$, if there is a collection $\sigma =(\sigma_1,\ldots,\sigma_p )$ of $n$-vectors $\sigma_i=(\sigma_i^1,\ldots,\sigma_i^n)\ (i=1,\ldots,p)$ with non-negative components such that either $$ \langle\sigma_i,\theta\rangle\equiv \sum_{j=1}^n\,\sigma_i^j\theta_j=1 \quad (i=1,\ldots,p) $$ or $$ \langle\sigma_i,\theta\rangle =-1 \quad (i=1,\ldots,p) $$ and the inequality $$ \langle\tau,{\bar u}\rangle >k\max\,\{{\bar u}_j: j=1,\ldots,n\} $$ holds for every vertex ${\bar u}=({\bar u}_1,\ldots,{\bar u}_n)$ of the convex polyhedral domain $D=D(\sigma )$ determined by the inequalities $$ u_j\ge 0 \quad (j=1,\ldots,n),\quad \langle\sigma_i,\theta\rangle\ge 1 \quad (i=1,\ldots,p). $$ Let $Q$ be an integer, $Q\ge 2$. The polynomial $$ P(x)= \bigl( \sum_{|\omega |=2}^Q\,P_{\omega}^1 x^{\omega},\ldots, \sum_{|\omega |=2}^Q\,P_{\omega}^m x^{\omega}\bigl) $$ is called $\tau$-{\it divisible} if $P_{\omega}^j\ne 0$ implies $\tau^1\le\omega^1,\ldots,\tau^n\le\omega^n$. \medskip {\bf Lemma 9.1.}\quad {\it Let $E$ and $F$ be finite-dimensional real vector spaces, $dim\,E=d$; ${\dot x}=Ax$ be a hyperbolic linear vector field on $E$ and ${\dot y}=By$ be a linear vector field on $F$. Let $\tau\in\Z_{+}^n$ satisfy condition $\a (k)$ with respect to the vector field ${\dot x}=Ax$. Let $P:E\to F$ be a $\tau$-divisible resonant polynomial. Let $Q_{\varepsilon}:E\times F\to F$ be a $C^{\infty}$ smooth family of resonant polynomials, $Q_{\varepsilon}(x,0)=0\ (x\in E)$, and $q_{\varepsilon}: E\to E$ be a $C^{\infty}$ smooth family of resonant polynomials. Then the extension $$ {\dot y}=By+Q_{\varepsilon}(x,y)+P(x), \quad {\dot x}=Ax+q_{\varepsilon}(x) \eqno {\rm (9.1)} $$ admits a $C^k$ smooth invariant section $y= \varphi_{\varepsilon}(x)$ which $C^{\infty}$ smoothly depends on the parameter $\varepsilon$.} \medskip The proof of this lemma is based on the techniques elaborated in [2], Chapter~II. It uses the iterative method of introducing additional variables (see Lemma~II.8.1) as well as Theorem~II.5.23 adapted to the case of vector fields. The details are left to the interested reader. Let us say that the hamiltonian $F$ of the vector field $\xi$ is reduced to the {\it simplified resonant polynomial $k$-normal form} if $F$ is a polynomial which contains only such resonant terms $x^{\tau}$ that $\tau$ does not satisfy condition $\a (k+1)$. \medskip {\bf Theorem 9.1.}\quad {\it Let the vector field $\xi$ and its hamiltonian $F$ satisfy the hypotheses of Theorem~8.1, then $F$ can be reduced to the simplified resonant polynomial $k$-normal form by a local canonical transformation of class $C^k$.} \medskip {\bf Proof.} Let $F$ be the resonant polynomial hamiltonian of the vector field $\xi$. Represent $F$ in the form $F=P+Q$, where $P$ is a $\tau$-divisible resonant polynomial and $\tau$ satisfies condition $\a (k+1)$. Let $A=D\xi (0)$ and $p$ be the polynomial vector field which corresponds to the hamiltonian $P$. Denote $F_{\varepsilon}=Q+\varepsilon P$, then $\xi_{\varepsilon}(x)=Ax+q(x)+\varepsilon p(x)$. Let us prove that the cohomology equations (5.1), (5.3), (5.5) with $G-F=P$ have $C^{k+1}$ smooth solutions. Consider the corresponding characteristic systems (5.2), (5.4) and (5.6). In each of the three cases, the characteristic system is of the form (9.1). According to Lemma~9.1, there exists a family of invariant sections $y=H_{\varepsilon}(x)$ of class $C^{k+1}$. The hamiltonian $H_{\varepsilon}$ determines a change of variables of class $C^k$ (see Section 4). Applying successively these arguments to all multi-indices $\tau$ satisfying condition $\a (k+1)$, we get the desired result. \begin{thebibliography}{99} \bibitem{1} Yu.S.Il'yashenko and S.Yu.Yakovenko. Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Uspekhi Mat.\ Nauk, 46, No.1 (1991), 3--39 (in Russian). \bibitem{2} I.U.Bronstein and A.Ya.Kopanskii. Smooth Invariant Manifolds and Normal Forms. World Scientific. 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