Let ($\mathcal{O}$, $\mathcal{M}$, k) be an DVR and $F_{1},...,F_{n} \in \mathcal{O}[X_{1},...,X_{n}]$ such that detJF = 1 where JF is the matrix $(\frac{\partial F_{i}}{\partial X_{j}})$. Suppose there exists $d_{1},...,d_{n} \in \mathcal{O}$ such that

$$u_{1}F_{1}(d_{1},...,d_{n})+\cdots+u_{n}F_{n}(d_{1},...,d_{n}) = 1$$

for some $u_{1},...,u_{n} \in \mathcal{O}$. In this case, we say that $F = (F_{1},...,F_{n})$ is an $n$-tuple unimodular. Let $G_{1},...,G_{n} \in \mathcal{O}[X_{1},...,X_{n}]$ defined by $G_{j} = F_{j}(F_{1},...,F_{n})$ for all $j$.

Is it $G = (G_{1},...,G_{n})$ be an $n$-tuple unimodular?