Let $n\geq 3$ be an integer. Let $f(x), g(x)$ be polynomials with real coefficiants such that the points $(f(1),g(1)), (f(2),g(2)),\cdots (f(n),g(n))$ in $\mathbb{R}^2$ form a regular $n$-gon in counterclockwise order. Prove that $\max(\deg f, \deg g)\geq n-1$.
I know that there is a solution which puts the polygon on the complex plane and setting $p(x)=f(x)+ig(x)$, then choosing suitable $a,b\in\mathbb{C}$ to scale $p(x)$ to have roots at $n$th roots of unity.
I was wondering if there is a solution without complex numbers.
Here is some progress: We know that $(f(k+1)-f(k))^2+(g(k+1)-g(k))^2=c$, for $1\leq k\leq n$ (with $n+1=1$). Then I set $p(x)=(f(x+1)-f(x))^2+(g(x+1)-g(x))^2-c$, which conviniently factors as $p(x)=q(x)(x-1)(x-2)\cdots(x-n)$. This gave me the bound $\max(\deg f, \deg g)\geq \frac{n+2}{2}$. I'm not sure if this helps.
