Let $n \geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials with real coefficients such that the points $(f(1), g(1)), (f(2), g(2)), \ldots, (f(n), g(n))$ in $\mathbb{R}^2$ are the vertices of a regular $n$-gon in counter-clockwise order. Prove that at least one of $f(x)$ and $g(x)$ has a degree greater than or equal to $n - 1$.
My solution
Let us assume $z = x + iy$ is a complex number. We define $h(z) = (z - 1)(z - \alpha_1)\ldots(z - \alpha_{n-1})$. $f(x)$ yields the abscissae of the vertices, and $g(x)$ yields the ordinates. So, $h(x, y) = (x + iy - 1)(x + iy - \alpha_1)\ldots(x + iy - \alpha_{n - 1})$. We can express $f(x) = \operatorname{Re}(h)$ and $g(x) = \operatorname{Im}(h)$ as the function definitions. From the definition of $h$, it is trivial to see that either $f$ or $g$ has a degree \geq to $n - 1$.
My doubt regarding the solution is its correctness, as well if correct then robustness. Can someone provide a clearer solution or improve upon this one? It would be helpful if it incorporates the n-th roots of unity.