If we put a regular polygon centered on the origin in $\mathbb{R}^2$ then we can think of $D_{2n}$ as isometries of the plane. What is the degree of the smallest polynomial invariant under these isometries?
I was thinking two, since we let $A$ be the matrix that rotates the plane by $2 \pi /n$, it will take $x\mapsto x \cos[(2 \pi)/n] + y \sin[(2 \pi)/n]$ and $y\mapsto y \cos[(2 \pi)/n] -x \sin[(2 \pi)/n] $ and if square these we get $(x \cos[(2 \pi)/n] + y \sin[(2 \pi)/n])^2+(y \cos[(2 \pi)/n] -x \sin[(2 \pi)/n])^2 = x^2+y^2$. So the polynomial is invariant under the generator for the rotations so it is invariant under all of them. Also if we let $B$ be the matrix that reflects about the $x$ axis then it is clearly invariant under this reflection too, so it must be invariant under all of $D_{2n}$....
Here are my two issues: How do I know there isn't a homogenous polynomial of two complex-variables of degree one. Also about those reflections...what if I had situated my polygon in such a way that the line of reflection had some weird slope...I couldn't figure out how check if $x^2+y^2$ was invariant under that.