Comments on generalized Heron polynomials and Robbins’ conjectures

Heron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, ( 4 A ) 2 is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223–236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523–530. MR 96k:51024] showed that for any cyclic polygon with n edges, ( 4 A ) 2 satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n = 5 and n = 6 . He conjectured the degree of this polynomial for all n , and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371–404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3–24. MR 2004d:51014]. A short independent proof will be shown here.