series $f \in {\mathbb C}[[x_0,x_1]]$ with $f(0)=0$. Carrying out a linear change of variables if necessary, we can assume that the curve $\{x_0=0\}$ is transversal to ${\mathcal C}$ and that the curve $\{x_1 = 0\}$ has maximal contact (among smooth curves) with ${\mathcal C}$. For $h \in {\mathbb C}[[x_0,x_1]]$, the *local intersection multiplicity* of ${\mathcal C}$ and the curve $\{h = 0\}$ is defined as $$(f,h)_0 := \dim_{\mathbb C}\frac{{\mathbb C}[[x_0,x_1]]}{(f,h)}.$$ This induces a valuation $$\nu_{\mathcal C}: \frac{{\mathbb C}[[x_0,x_1]]}{(f)} \setminus \{0\} \longrightarrow {\mathbb N}: h \mapsto (f,h)_0.$$ The image of this valuation is called the *semigroup of ${\mathcal C}$* and denoted by $\Gamma({\mathcal C})$. Because ${\mathbb N}\setminus \Gamma({\mathcal C})$ is finite, there exists a unique minimal system of generators $(\bar{\beta}_0,\ldots,\bar{\beta}_g)$ of $\Gamma({\mathcal C})$ satisfying ${\bar{\beta}}_0 < \cdots < {\bar{\beta}}_g$ and $\gcd({\bar{\beta}}_0, \ldots, {\bar{\beta}}_g) = 1$, see for instance [@Z]. Additionally, we introduce the integers $e_i:=\gcd(\bar{\beta}_0,\ldots,\bar{\beta}_i)$ for $i=0,\ldots,g$ and $n_i:=\frac{e_{i-1}}{e_i}$ for $i=1,\ldots,g.$ From the minimality of the generators $(\bar{\beta}_0,\ldots,\bar{\beta}_g)$, one can easily see that ${{\bar{\beta}}_0 = e_0 > e_1 > \cdots > e_g = 1}$ and that $n_i \geq 2$ for all $i = 1,\ldots, g$. One can also show that every $n_i\bar{\beta}_i$ for $i=1,\ldots,g$ is contained in the semigroup generated by 