Binary forms, equiangular polygons and harmonic measure

Consider binary forms F having complex coefficients and discriminant D_F ≠ 0. In a sequence of previous papers, the first author studied the area AF of the planar region |F(x, y)| ≤ 1 defined by forms F of this type in connection with the enumeration of integer lattice points. In particular, the first author showed that the GL₂(R)-invariant quantity |D_F|^1/n(n-1)A_F is uniformly bounded when the degree of F is at least three, and conjectured that this quantity is maximized over the forms of degree n by forms with a complete factorization over R and n equally spaced asymptotes. The first author obtained his results using a standard integral representation of AF over the real line. In this paper we establish GL₂(C) invarianc ε of |D_F|^1/n(n-1)A_F with respect to Lagrangian planes in C² (a fact not previously noticed by many earlier authors), and we subsequently give integral representations of AF over every circle in the complex plane. In particular, we give a representation over the unit circle and we use this representation to give an explicit formula in terms of Beta functions for the conjectured maximum value of |D_F|^1/n(n-1)A_F. It turns out that integration over the unit circle is directly linked to binary forms in the complex indeterminates ¯z and z. In addition, we reformulate the maximization problem for binary forms in purely geometricand potential theoretic terms as a maximization problem for harmonicmeasu res on the edges of equiangular polygons, with the inner harmonicradius of the polygon being normalized. In this context the conjectured extremal polygon is the regular n-gon. We conclude the paper with tables that summarize the many equivalent formulas for A_F, D_F and related quantities.