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 $\mathbb{C}$



Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
This algorithm is called the Newton Polygon, and it was really invented and carefuly
described by Newton, with examples, see for example,


MR1836037 Fischer, Gerd Plane algebraic curves. Translated from the 1994 German original by Leslie Kay. Student Mathematical Library, 15. American Mathematical Society, Providence, RI, 2001. 


or any other book with a similar title, or Mathematical Papers of Newton himself. 


In XIX century this algorithm was a part of high school curriculum, at least in England, see, for exapmple


JFM 31.0103.05 Chrystal, G.
Algebra. Elementary textbook. 2. edition. Part II. (English)
London. 632 S. Published: 1900


I think, for this reason, those who wrote university algebra textbooks in 
XX century did not care to include it (Lang, Van der Waerden,...).


