\[
  A_n = \mathrm{Im}(\mathbf{Z} \to S_n)
\]

\[
  A_n = \mathrm{Im}(P \to S_n)
\]

\[
  A_n = P[t_1, \ldots, t_n]
\]

\[
  A_n = P[t_1, \ldots, t_n]_{\mathfrak{m}_n}, \qquad
  \mathfrak{m}_n = \sum t_i\, P[t_1, \ldots, t_n].
\]

\[
  \begin{cases}
    \text{a}_1)\ 1 \in A_0 \\
    \text{a}_2)\ \text{le morphisme « différence » } X_1 \times X_1 \to X_1
      \text{ est } A_*\text{-morph.} \\
    \text{a}_3)\ \text{le morphisme « produit » } X_1 \times X_1 \to X_1
      \text{ est } A_*\text{-morph.}
  \end{cases}
\]

\[
  A_n = P\{t_1, \ldots, t_n\} \qquad
  \bigl(\text{clôture hensélienne de } P[t_1, \ldots, t_n]_{\mathfrak{m}_n}\bigr)
\]

\[
  (t, x) \mapsto (\exp t\xi) . x \;:\; \underline{X_1} \times X \to X
\]

\[
      \frac{d}{dt}\, x(t) = \xi(t, x)
    \]

\[
  \mathcal{L}(k_0) \xrightarrow{\text{Hausdorff}} \mathrm{G\text{-}f}(k_0)
  \xrightarrow{\text{ext.\ de la base}} \mathrm{G\text{-}f}(k)
\]

\[
  \overline{W}(\mathfrak{g}_0) \times \overline{W}(\mathfrak{g}_0) \to
  \overline{W}(\mathfrak{g}_0)
\]

\[
  \mathfrak{P} \subset \Theta(X)
\]

\[
      \mathbf{T}(Y)_0 \xrightarrow{\mathbf{T}(i)_0} \mathbf{T}(X)_0 \to
      \mathbf{T}(X)_0 / \mathfrak{P}(0)
    \]

\[
      \pi : X \to Y
    \]

\[
  \mathbf{T}(Z)_0 \xrightarrow{\mathbf{T}(i)_0} \mathbf{T}(X)_0 \to
  \mathbf{T}(X)_0 / \mathfrak{P}(0)
\]

\[
  X \xrightarrow{\pi} Z
\]

\[
  A_n \subset S_n
\]

\[
  p^*(t_i) = f_i \in S_m \text{ est } \in A_m \text{ pour tout } 1 \leq i \leq n .
\]

\[
  A_n = \mathrm{Im}\bigl(\mathbf{Z}[t_1, \ldots, t_n] \to k[[t_1, \ldots, t_n]]\bigr)
  \simeq \Lambda[t_1, \ldots, t_n]
\]

\[
  \Lambda = \mathrm{Im}(\mathbf{Z} \to k)
  \begin{cases}
    \simeq \mathbf{Z} & \text{ou} \\
    \simeq \mathbf{Z}/n\mathbf{Z} & \text{pour } n \in \mathbf{N}^* \text{ convenable.}
  \end{cases}
\]

\[
      \varepsilon\Bigl(\det\Bigl(\frac{\partial f_i}{\partial t_j}\Bigr)\Bigr)
      = \det\Bigl(\varepsilon\Bigl(\frac{\partial f_i}{\partial t_j}\Bigr)\Bigr)
      = \det\Bigl(\frac{\partial f_i}{\partial t_j}(0)\Bigr)
    \]

\[
  u : (x_1, \ldots, x_n, x_{n+1}) \mapsto
  (x_1, \ldots, x_n, f(x_1, \ldots, x_n) . x_{n+1})
\]

\[
  (x_1, \ldots, x_n, x_{n+1}) \mapsto
  (x_1, \ldots, x_n, f(x_1, \ldots, x_n)^{-1} x_{n+1}),
\]

\[
  P^\varepsilon(\zeta) = \zeta^N + a_1 \zeta^{N-1} + \cdots + a_N = 0 ,
  \quad \text{où } a_i = \varepsilon(f_i) = f_i(0) \in k_0 ,
\]

\[
  G = QF + \sum_{0 \leq i \leq N-1} b_i Z^i \qquad
  Q \in A[[Z]],\ b_i \in A
\]

\[
  \dot{a}_i = 0 \quad \text{pour } 0 \leq i \leq N-1
\]

\[
  \dot{F} = \dot{a}_N Z^N + \cdots = \dot{a}_N Z^N (1 + \cdots)
\]

\[
  A_0[[Z]]/\dot{F} \simeq A_0[[Z]]/Z^N
\]

\[
  \begin{array}{rccc}
    \varphi : & A^N + A[[Z]] & \longrightarrow & A[[Z]] \\
    & \bigl((b_i)_{0 \leq i \leq N-1}, Q\bigr) & \longmapsto &
      \sum_{0 \leq i \leq N-1} b_i Z^i + QF
  \end{array}
\]

\[
  (\mathrm{Gr}\, A)^N \times (\mathrm{Gr}\, A)[[Z]] \to (\mathrm{Gr}\, A)[[Z]]
\]

\[
  \bigl((b_i), Q\bigr) \longmapsto \sum_{0 \leq i \leq N-1} b_i Z^i + Q \dot{F}
\]