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 $f_N$?
  • Consider the branching around every finite singular point and the region of convergence of the power series representations of these branches centered at the corresponding singular points. Do these series completely represent the function, except at the singular points, or are there regions in the $z$-plane missed by all series?

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