Explicit plane models for all Galois quotients of the Hermitian curve

Let $H=k(x,y)$ be the Hermitian function field $y^q+y=x^{q+1}$ over $k=\mathbb{F}_{q^2}$, and let $G$ be a subgroup of its $k$-automorphism group $\operatorname{PGU}(3,q)$. We give a uniform construction of a plane model for $H^G$. For each of $x$ and $y$, we multiply its distinct translates under the stabilizer $S$ in $G$ of their common pole, then sum the translates of that product over $G/S$. The two resulting functions $U,V$ generate $H^G$. In $H^G$, their only pole is the restriction of the common pole, with coprime orders $a=q/|G_x|$ and $b=(q+1)/|G_y|$, where $G_x,G_y$ are the subgroups of $G$ fixing $x,y$, respectively. Their plane equation has degree $\max(a,b)\leq q+1$ and can be recovered from finitely many Laurent coefficients of $U,V$ at the common pole of $x,y$. All steps are effective given matrix generators of $G$. The total delta invariant of this affine model is $g(H^{G_xG_y})-g(H^G)=\frac{1}{2}(a-1)(b-1)-g(H^G)$. Consequently $g(H^G)\leq\frac{1}{2}(a-1)(b-1)$; when $g(H^G)\geq2$, the model is smooth if and only if $G=G_xG_y$. The generator construction also applies to other function fields admitting two functions whose only pole is a common rational place, with coprime orders.