# How many distinct ways can an algebraic function fully-ramify?

1. Oct 17, 2012

### jackmell

Take for example, the function:

$$a_0(z)+a_1(z)w+a_2(z)w^2+a_3(z) w^3+a_4(z)w^4+a_5(z)w^5=0$$

with the degree of $a_n(z)$ also five.

What are necessary or sufficient conditions imposed on the polynomials $a_n(z)$ to cause $w(z)$ to fully-ramify at the origin?

I can easily think of one sufficient condition:

$$z-w^5=0$$

Can we do better than that?

Also, can anyone give me a reference where this type of problem may be addressed? I do have several texts on algebraic functions but can't find anything about it.

Thanks,
Jack

Last edited: Oct 17, 2012