Algebraic Relations Among Mean Value Coordinates: A Complete Answer for Quadrilaterals and Partial Results for Larger Polygons

Mean value coordinates of a planar polygon P with n vertices define a map from P to P^(n−1). At an Oberwolfach mini-workshop in 2022, F. Sottile asked for the homogeneous equations of the Zariski closure S_P of its image. We answer this question for quadrilaterals and give partial results for larger polygons. For every quadrilateral with no three vertices collinear, the ideal of S_P is generated by one explicit irreducible polynomial F_P of degree 14. It is obtained by clearing the square roots in the relation λ_1|x − v_1| − λ_2|x − v_2| + λ_3|x − v_3| − λ_4|x − v_4| = 0, which follows at once from Floater's formula; for the unit square, F_P/16 has 116 integer terms. For every n ≥ 4, S_P is an irreducible surface, and the projection w ↦ Σ w_i v_i / Σ w_i restricts to a map of degree 2^(n−1) from S_P to the plane. Over a general point x, a point w with Σ w_i ≠ 0 and Σ w_i (v_i − x) = 0 lies on S_P exactly when the closed polygon with edge vectors w_i J(v_i − x), J a quarter turn, is circumscribed about a circle. The passage from coordinates to circumscribed polygons is the classical tangent-length picture behind Floater's construction; its converse is what produces equations. For n = 5, 6, 7, computations modulo primes for sample polygons give relations of lowest degree 10, 6, 7, and deg S_P = 52 for n = 5; for a single hexagon they give deg S_P = 152. These values are consistent with a conjectured value 2^(n−3)(6n − 17). A generating set for n ≥ 5 remains open. This is an unrefereed note.