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

[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

 

[jvicens]

Hello Jack,

Thank you for bringing up this interesting question about the necessary and sufficient conditions for a function to fully-ramify at the origin. To fully understand this problem, we first need to define what it means for a function to "fully-ramify". A function is said to fully-ramify at a point if all of its roots are equal to that point. In other words, the function has a repeated root at that point.

Now, let's consider the given 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

If we want this function to fully-ramify at the origin, we need to find conditions on the polynomials a_n(z) such that all of the roots of the function are equal to the origin. One sufficient condition for this is the one you mentioned: z-w^5=0. However, this is not the only condition that can cause full-ramification at the origin.

In fact, there are infinitely many conditions that can cause full-ramification at the origin. For example, any function of the form z-w^n=0, where n is any positive integer, will fully-ramify at the origin. This is because all of the roots of this function will be equal to the origin.

To find a more general and comprehensive set of conditions, we need to look at the properties of the polynomials a_n(z). One necessary condition for full-ramification at the origin is that the polynomial a_n(z) must have a root at the origin for all n. This ensures that all of the terms in the given function have a common factor of w and will therefore have a repeated root at the origin.

Another necessary condition is that the sum of the degrees of the polynomials a_n(z) must be equal to the degree of the function, which in this case is 5. This is because if the sum of the degrees is less than 5, there will be terms in the function that do not have a root at the origin, and therefore the function will not fully-ramify.

As for references, this type of problem falls under the study of algebraic functions and their properties. I would recommend looking into books on complex analysis and algebraic geometry for more information on this topic.

I hope this helps answer your