Can ramification geometry of algebraic function be computed algebraically?

[jackmell]

Suppose I'm given a random function:

$$(-8+5 z+4 z^2)\text{}+(7 z+6 z^4-7 z^5)w+(3 z^2-z^3)w^2+(-8 z-2 z^4-2 z^5)w^3+(3-4 z+4 z^2+7 z^3+6 z^4-8 z^5)w^4+(-6 z+4 z^4)w^5=0$$

Is there no way to determine it's ramification geometry at each singular (critical) point algebraically? I'm pretty sure the multiplicity of the zeros at the critical points do not determine this geometry.

Just seem there should be an algebraic way to do this or is there some theorem which deems this impossible?

Also, I'm not sure if there are several definitions of "singular" point of algebraic function. The one I'm using is that for the function:

$$f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n=0$$

the point $z_s$ is a singular point when either a_n(z_s)=0 or the discriminant of f at $z_s$ is zero.

Thanks,
Jack

 

[jvicens]

Dear Jack,

Thank you for your post. I can understand your curiosity about determining the ramification geometry at each singular point of a given function. However, I must say that there is no general algebraic method to do so.

Firstly, let me clarify that there are indeed multiple definitions of "singular" points for algebraic functions. The one you have mentioned is one of the definitions, but there are others as well. Therefore, it is important to specify which definition we are using when discussing singular points.

Now, coming to your main question about determining ramification geometry at singular points algebraically, I'm afraid there is no theorem or method that can guarantee this. The multiplicity of zeros at critical points can give you some information about the function, but it is not enough to determine the entire ramification geometry.

The reason for this is that ramification geometry is a complex and intricate concept, and it involves not only the zeros and multiplicities of a function, but also the behavior of the function in the neighborhood of those points. This behavior is highly dependent on the specific function and cannot be captured by a general algebraic method.

In some special cases, it may be possible to determine the ramification geometry algebraically, but it is not a general rule. In most cases, it requires a combination of analytical and numerical methods to fully understand the ramification geometry of a function.

I hope this clarifies your doubts. Let me know if you have any further questions.