Let $\Phi (z,t)$ be a polynomial given by $$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$ Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation $$ \Phi(z(t), t) =0 $$ that is close to zero, can be expressed as a formal power series in $t^{1/r}$ for some positive integer $r$. Moreover, this formal power series has a non zero radius of convergence. This follows from the "Newton Pusieux Theorem".

My question is the following: Is there some procedure/algorithm to find out what this $r$ is? Of course it will depend on the $A_i$.

As an example, suppose we want solutions for $$ \Phi(z,t) = z^2 + K z + t =0$$ where $K$ is some constant. Then a solution $z(t)$ is a power series in $t$ if $K \neq 0$. It is a power series in $t^{1/2}$ if $K=0$.

In general is there some algorithm to find this $r$?

Everything is over the field of complex numbers $\mathbb{C}$