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Extremely Weird Algebra

Source: 2023 RMM, Problem 3

March 1, 2023
algebrapolynomialanalytic geometryRMM 2023

Problem Statement

Let n≥2n\geq 2 be an integer and let ff be a 4n4n-variable polynomial with real coefficients. Assume that, for any 2n2n points (x1,y1),…,(x2n,y2n)(x_1,y_1),\dots,(x_{2n},y_{2n}) in the Cartesian plane, f(x1,y1,…,x2n,y2n)=0f(x_1,y_1,\dots,x_{2n},y_{2n})=0 if and only if the points form the vertices of a regular 2n2n-gon in some order, or are all equal.
Determine the smallest possible degree of ff.
(Note, for example, that the degree of the polynomial g(x,y)=4x3y4+yx+x−2g(x,y)=4x^3y^4+yx+x-2 is 77 because 7=3+47=3+4.)
Ankan Bhattacharya
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