Non-degeneracy conditions for plane branches

Let $x=t^n$, $y={\sum }_{i=1}^{\mathrm{\infty }}{a}_i{t}^i$ be a parametrisation of the germ of a complex plane analytic curve $\mathrm{\Gamma }$ at the origin. Then $\mathrm{\Gamma }$ has the implicit equation $f(x,y)=0$ in the neighbourhood of the origin, where $f=\sum c_{ij}{x}^i{y}^j$ is a Weierstrass polynomial in $\mathbb{C}[[x]][y]$ of degree $n$. Every polynomial depending on coefficients $c_{ij}$ can be expressed as a polynomial depending on the coefficients $a_i$.