https://ift.tt/2Z4Rs1w A Belyi-extender (or dessinflateur ) $\beta$ of degree $d$ is a quotient of two polynomials with rational coefficients \[ \beta(t) = \frac{f(t)}{g(t)} \] with the special properties that for each complex number $c$ the polynomial equation of degree $d$ in $t$ \[ f(t)-c g(t)=0 \] has $d$ distinct solutions, except perhaps for $c=0$ or $c=1$, and, in addition, we have that \[ \beta(0),\beta(1),\beta(\infty) \in \{ 0,1,\infty \} \] Let’s take for instance the power maps $\beta_n(t)=t^n$. For every $c$ the degree $n$ polynomial $t^n – c = 0$ has exactly $n$ distinct solutions, except for $c=0$, when there is just one. And, clearly we have that $0^n=0$, $1^n=1$ and $\infty^n=\infty$. So, $\beta_n$ is a Belyi-extender of degree $n$. A cute observation being that if $\beta$ is a Belyi-extender of degree $d$, and $\beta’$ is an extender of degree $d’$, then $\beta \circ \beta’$ is again a Belyi-extender, this time of degree $d.d’$. That is, Belyi-extenders ...
