### Unsolved problems

$$\newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{Log}{\text{Log}} \newcommand{Arg}{\text{Arg}} \newcommand{pLog}{\text{pLog}}$$
1. 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 $f_N$?
2. Can an algorithm similar to Newton Polygon be devised to compute annular Puiseux series?
3. Can a folded version of Buchberger's algorithm be derived? See Folded polynomial systems. for background on folded systems.

Consider the generic quadratic polynomial system $f(x,y)\in\mathbb{Q}[x,y]$: $$f(x,y)=\left\{\begin{array}{c} a_0+a_1x+a_2y+a_3x^2+a_4y^2+a_5xy\\ b_0+b_1x+b_2y+b_3x^2+b_4y^2+b_5xy \end{array}\right\}$$ with random $a_i,b_i\neq 0$ and absolute value less than one . A fixed-point $(x_f,y_f)$ is a point such that $f(x_f,y_f)=(x_f,y_f)$. Next consider a fixed point of the $n$-folded system: $$f^{((n))}(x_f,y_f)=\underbrace{f(f(f(\cdots f))\cdots)}_{\text{n-times}}=(x_f,y_f)$$ This is equivalent to solving for the roots of $V_{n}=F^{((n))}-\{x,y\}$. Computing a Groebner basis for $V_n$ directly using Buchberger's algorithm for anything but small systems is practically inaccessible computationally. Is is possible to derive a more efficient version of Buchberger's algorithm that takes advantage of the folded properties of $f^{((n))}$? That is, can a folded version of Buchberger's algorithm be devised to improve the computation speed of computing a Groebner basis for $V_n$?

4. If a system $f(z,w)=\{g(z,w),h(z,w)\}-\{z,w\}$ has finite dimension, that is to say it has a finite number of zeros, under what conditions is the folded system $f^{((n))}-\{z,w\}$ also finite dimensional?

5. Are the roots of a finite-dimensional generic system $V_n$ bounded?