Question

Hi everyone. While looking for examples of asymptotically bad towers of function fields over a finite field $F_q$ defined by a Kummer equation of the form $y^m=f(x)$ with $p\equiv 1\mod m$ where $p=Char(F_q)$ the following situation came up: let $$a, b, c, d\in F_q$$ such that $a$, $c$ and $ad-bc$ are non zero in $F_q$. Now consider the rational function $$g(x)=(a/c)^m-\left(\frac{ax+b}{cx+d}\right)^m$$ and let $f(x)$ be the denominator of $g(x)$. Clearly $f(x)$ is a polynomial of degree $m-1$ with coefficients in $F_q$. To my surprise, after many numerical experiments, $f(x)$ seems to be a separable polynomial splitting into linear factors in $F_q[x]$. I could prove this in one particular situation. Maybe someone already knows this is true or, equally better, someone already found a counterexample. Thanks!