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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.

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    I cannot understand what you are asking really... Have you tried writing down a general polynomial of degree $1$, seeing how it changes under the elements of your group, and finding which are fixed by the whole group?2012-04-23

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Second question first - $x^2+y^2$ is just the distance of the point from the origin, and that is invariant under rotations and also under reflection in any line thrugh the origin, so it doesn't matter how you situate your polygon.

For the first question, imagine the polygon situated so there's a vertex at $(1,0)$. Then one of the reflections is complex conjugation. That will take any linear $ax+by$ to $ax-by$, and those can only be equal if $b=0$, so your polynomial must be $ax$. But that's clearly not invariant under rotation (if $a\ne0$).

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    @Mariano, true enough. I just wrote it down the way I visualized it.2012-04-23
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This should not be taken too seriously...:

You can use Molien's formula to find the dimension of the space of invariant polynomials of all degrees and, in particular, of degree one.