Question 6
Let $n \ge 2$ be an integer, and let $f$ be a $4n$-variable polynomial with real coefficients, such that, for any $2n$ points $(x_1, y_1), \dots, (x_{2n}, y_{2n})$ in the Cartesian plane,
$$
f(x_1, y_1, \dots, x_{2n}, y_{2n}) = 0
$$
if and only if they form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$.
$$
f(x_1, y_1, \dots, x_{2n}, y_{2n}) = 0
$$
if and only if they form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$.