### Unsolved problems

$$\newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}}$$

The reader is asked to review the sections in sequential order as each succeeding section builds on a previous section.

# Puiseux Series and the Newton Polygon Algorithm

## Unsolved problems

$$\newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{\wvalx}{\underbrace{z^{\lambda_4}(c_4+w_5)}_{w_4}} \newcommand{wvalxx}{\underbrace{z^{\lambda_3}(c_3+\wvalx)}_{w_3}} \newcommand{wvalxxx}{\underbrace{z^{\lambda_2}\{c_2+\wvalxx\}}_{w_2}} \newcommand{wvalxxxx}{z^{\lambda_1}\big(c_1+\wvalxxx\big)}$$
• Consider the family of algebraic functions: $$f_N(z,w)=z-\prod_{j=1}^{N}(w-j)^j=0,\quad N\geq 2$$ and their associated set of power expansions $\left\{w_d(z)\right\}_{d=1}^{N}$ with $w_d(z)=\displaystyle\sum_{n=0}^{\infty} c_n \left(z^{1/d}\right)^n.$ The $w_1(z)$ branches for members up to $N=10$ have radii of convergences extending out to the most distant singular point of $f_N$. Is this the case for all $f_N$?
• Can an algorithm similar to Newton Polygon be devised to compute annular Puiseux series?