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Polygon vanishing locus of polynomial

Source: The 1st India-Iran Friendly Competition Problem 3

June 12, 2024
algebraPolynomialspolynomial

Problem Statement

Let n≥3n \ge 3 be an integer. Let P\mathcal{P} denote the set of vertices of a regular nn-gon on the plane. A polynomial f(x,y)f(x, y) of two variables with real coefficients is called <spanclass=′latex−italic′>regular</span><span class='latex-italic'>regular</span> if P={(u,v)∈R2 ∣ f(u,v)=0}.\mathcal{P} = \{(u, v) \in \mathbb{R}^2 \, | \, f(u, v) = 0 \}. Find the smallest possible value of the degree of a regular polynomial.
Proposed by Navid Safaei
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