= Binary dihedral invariant hypersurface
{title2=$W^2-VU^2+4V^{n+1}=0$}
Let $\zeta=e^{\pi i/n}$ and let the <binary dihedral group> act on $\mathbb C^2$ by $(x,y)\mapsto(\zeta x,\zeta^{-1}y)$ and $(x,y)\mapsto(-y,x)$. For $n\ge2$, the invariants
$$
U=x^{2n}+y^{2n},\qquad V=x^2y^2,\qquad W=xy(x^{2n}-y^{2n})
$$
present the invariant ring as $\mathbb C[U,V,W]/(W^2-VU^2+4V^{n+1})$. First take the cyclic invariants $A=x^{2n},B=y^{2n},C=xy$ with $AB=C^{2n}$. The residual involution interchanges $A,B$ and negates $C$, so invariant normal forms are polynomials in $A+B,C^2$ plus $C(A-B)$ times such polynomials. For $n=3$ the group has order twelve and the equation is $W^2-VU^2+4V^4=0$.
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