Proposition 3

Let $u = \sum a_\nu X^\nu \in \mathbf{A}[[\mathbf{I}]]$ and $m$ an integer $> 0$. For every $n \in \mathbf{N}$ let $(\alpha_{\nu, n})_{\nu \in \mathbf{N}^{(I)}}$ be the family of coefficients of $u^n$. If $\alpha_0^m = 0$, then $\alpha_{\nu, n} = 0$ for $n \geq |\nu| + m$.

Let $\nu \in \mathbf{N}^{(I)}$ and $n \in \mathbf{N}$. We have

If $n \geq |\nu| + m$ and $\nu(1) + \cdots + \nu(n) = v$, we have $|\nu(1)| + \cdots + |\nu(n)| \leq n - m$. We thus have $v(r) = 0$ and so $\alpha_{v(r)} = a_r$, for at least $m$ distinct values of $r$; it follows that $\alpha_{\nu(1)} \cdots \alpha_{\nu(n)} = 0$, whence the result.