Minimal-polynomial criterion for tree-pair representations of algebraic Bieri–Strebel groups

Let $\textbf{F}_{\beta}$ be an algebraic Bieri–Strebel group with associated subdivision polynomial $P(x)$, where $\beta$ is the relevant positive root. Consider polynomials of the form

$$a_nx^n+\cdots+a_1x-1,\qquad a_i\in\mathbb{Z}_{\geq 0}.$$

Minimal-polynomial conjecture. A well-defined tree pair for $\textbf{F}_{\beta}$ can be derived from $P(x)$ only if $P(x)$ is the minimal polynomial having root $\beta$ among polynomials of this form.

This conjecture proposes a necessary condition for the tree-pair construction to be well defined, motivated by examples in which nonminimal or transformed subdivision polynomials produce breakpoint sets or interval subdivisions incompatible with the intended Bieri–Strebel group. The source gives no resolution, so the conjecture remains open.