7 problems

Let $G$ be a group, and let $\{w_i=1\}_{i\in I}$ be a system of equations in variables $\{x_j\}_{j\in J}$ over $G$. For each equation, form the row of exponent sums of the variable…

Kervaire–Laudenbach–Howie conjecture. Every independent system of equations over $G$ has a solution in an overgroup of $G$.

An equation $w(x)=1$ over a group $G$, where $w(x)\in G\ast\langle x\rangle_\infty$, is unimodular if the exponent sum of $x$ in $w(x)$ equals $\pm1$. It is solvable if there exist…

Let $G$ be a group, and let $\mathcal{W}$ be a finite system of $m$ equations in variables $x_1,\dots,x_n$ over $G$. Let $M$ be the $m\times n$ matrix whose $(i,j)$-th entry is the…

Torsion-free solvability conjecture. Every such equation is solvable over $G$, including when its content is trivial.

The nonsingular equation conjecture. If $w$ is non-singular, then it is solvable over $G$. If $G$ is finite, then a solution can be found in a finite extension.

Finite parametric-word description conjecture. The solution set of any system of equations could be described by a finite system of parametric words.