Generalization of Beardon’s theorem to tame rational functions

Let $\mathbb{K}$ be a field of characteristic $\operatorname{char}(\mathbb{K})$, and let $f$ be a rational function over $\mathbb{K}$ whose degree is not a multiple of $\operatorname{char}(\mathbb{K})$. For a rational function $g$, write $\Gamma(g)$ for its group of compositional symmetries. The preceding theorem asserts that for tame polynomials $p_1,\ldots,p_m$, if $k_1,\ldots,k_m,k$ are the orders of $\Gamma(p_1),\ldots,\Gamma(p_m),\Gamma(p_1\circ\cdots\circ p_m)$, respectively, then $k$ divides $k_1\cdots k_m$. Generalized Beardon conjecture. Theorem is true for every rational function whose degree is not a multiple of the characteristic of the field. The conjecture would extend the divisibility result from tame polynomials to rational functions; the preceding example shows that the polynomial proof does not directly generalize in the rational case because fields of the same degree need not be unique.