Polynomial-time approximate-root conjecture for circuit-system Gale duals

Let $n\in\mathbb{N}$), let $\gamma_{i,j}\in\mathbb{Q}$ and $b_i\in\mathbb{Z}$ for all $i,j$, and consider the univariate rational function

$$g(u):=\prod_{i=1}^{n+1}\left(\gamma_{i,1}u+\gamma_{i,0}\right)^{b_i}-1.$$

For coprime integers $u,v$, define the logarithmic height of $u/v$ by $h(u/v):=\max\{|u|,|v|\}$, with $h(0):=0$. Set $B:=\max_i|b_i|$ and $\log H:=\max_{i,j}h(\gamma_{i,j})$. An approximate root is understood in the sense of Smale.

Polynomial-time approximate-root conjecture. Following this notation, one can find approximate roots for all the real roots of $g$ in time polynomial in $\log^n(BH)$.

This conjecture concerns the real-solving complexity of circuit systems, which generalize univariate trinomial equations. It is motivated by reductions of circuit-system solving to real roots of Gale-dual rational functions; its status is not resolved in the supplied text.