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

- Home
- Introduction
- Section 0: Preliminaries
- Section 1: Plotting Algebraic Functions
- Section 2: An Improved Plotting Method
- Section 3: Applying Laurent's Theorem to Algebraic Functions
- Section 4: Applying the Residue Theorem to Algebraic Functions
- Section 11: Riemann Surfaces
- Section 12: Evaluating the Indeterminant Form

- Puiseux Series
- AFRender
- Mathematica Code
- Useful links
- Unsolved problems

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

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

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