Umemura's Theorem on Solving any Algebraic Polynomial Equation

Question

I have recently read Bruce King's Beyond the Quartic Equation. In the last pages, he shows a theorem, Umemura's Theorem, which shows how to solve any polynomial equation. I have tried to understand the concepts behind, but failed. Then, I started searching everywhere I could and found these things:

• This is the only question (no answered by the way) I have seen in Stack about this: https://mathoverflow.net/questions/151498/understanding-umemuras-theorem-for-roots-of-algebraic-equations
• Something related: https://en.wikipedia.org/wiki/Thomae's_formula
• And also Wiki pages about the topics mentioned bellow

As far as I went, it seems I need to understand what are:

1. Elliptic Functions/Integrals
2. Theta Functions
3. Modular Functions
4. Period Matrix
5. Theta Functions with those matrices filled up with zeros and ones (shown in the Thomae's formula link)
6. Higher Genus Theta Functions

I tried to read the wiki pages about these topics above, however, sometimes, I do not get a thing. Well, I'm here to ask your opinion on what to do, what and how should I attack first, since I only know Single-Variable, Multi-Variable and Vectorial Calculus, some ways of solving ODEs, Linear Algebra and that's it.

Of course, if you know any of these concepts and share with me or at least recommend something "easy" to read, I would really appreciate.

Thanks