Terminology
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In this section, we present some notions and terminology that will be used throughout the paper. In particular, we need some previous results concerning local parametrizations and Puiseux series. For further details see [@Duval89], Section 2.5 in [@SWP], [@Stad00], and Chapter 4 (Section 2) in [@Walker].

We denote by ${\Bbb C}[[t]]$ the domain of [*formal power series*]{} in the indeterminate $t$ with coefficients in the field ${\Bbb C}$, i.e. the set of all sums of the form $\sum_{i=0}^\infty a_it^i$, $a_i \in {\Bbb C}$. The quotient field of ${\Bbb C}[[t]]$ is called the field of [*formal Laurent series*]{}, and it is denoted by ${\Bbb C}((t))$. It is well known that every non-zero formal Laurent series $A \in {\Bbb C}((t))$ can be written in the form $A(t) = t^k\cdot(a_0+a_1t+a_2t^2+\cdots), {\rm\ where\ }
    a_0\not= 0 {\rm\ and\ } k\in \Bbb Z.$ In addition, the field ${{\Bbb C}\ll t\gg}:= \bigcup_{n=1}^\infty {\Bbb C}((t^{1/n}))$ is called the field of [*formal Puiseux series*]{}. Note that Puiseux series are power series with fractional exponents. In addition, every Puiseux series, $\varphi$, has a bound for the denominators of exponents with non-vanishing coefficients, which is known as [*the ramification index*]{} of the series. We denote it as $\nu(\varphi)$ 