for these and other properties of $Y$.

We can construct $Y$ as a deformation of ${\mathcal C}$ as follows. First of all, we can consider a *system of approximate roots* or a *minimal generating sequence* $(x_0, \ldots, x_g)$ of the valuation $\nu_{{\mathbb C}}$, which consists of elements $x_i \in {\mathbb C}[[x_0,x_1]]$ for $i=0,\ldots,g$ such that $\nu_{{\mathbb C}}(x_i)=\bar{\beta}_i$. For $i=0,1$, this condition is equivalent to the above assumptions on $x_0$ and $x_1$, respectively. These elements satisfy equations of the form $$x_{i+1}=x_i^{n_i}-c_ix_0^{b_{i0}}\cdots x_{i-1}^{b_{i(i-1)}}- \sum_{\gamma=(\gamma_0,\ldots,\gamma_i)} c_{i,\gamma }x_0^{\gamma_0}\cdots x_i^{\gamma_i}, \qquad  i = 0, \ldots, g,$$ where $x_{g+1}=0, c_i \in {\mathbb C}\setminus \{0\},  c_{i,\gamma }\in {\mathbb C},$ $0\leq \gamma_j <n_j$ for $1\leq j \leq i$, and $\sum_{j=0}^i \gamma_j\bar{\beta}_{j}>n_i\bar{\beta}_{i}.$ For more details, we refer to [@AM], [@Sp] and [@T1]. Note that these equations realize ${\mathcal C}$ as a complete intersection in $({\mathbb C}^{g+1},0)$. Even more, this complete intersection is Newton non-degenerate in the sense of [@AGS] and [@Te1]. It was proven (resp. conjectured) that such an embedding always exists in characteristic $0$ [@Te2] (resp. in positive characteristic [@T2]). We now consider the following slight modification of the above equations in the variables $x_0,\ldots, x_g$ including an extra variable $v$: $$vx_{i+1}=x_i^{n_i}-c_ix_0^{b_{i0}}\cdots x_{i-1}^{b_{i(i-1)}}- \sum_{\gamma=(\gamma_0,\ldots,\gamma_i)} c_{i,\gamma }vx_0^{\gamma_0}\cdots x_i^{\gamma_i}, \qquad  