Binary forms, equiangular polygons and harmonic measure

Michael A. Bean, Richard S. Laugesen

Research output: Contribution to journalArticlepeer-review


Consider binary forms F having complex coefficients and discriminant DF ≠ 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 |DF|1/n(n-1)AF 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 |DF|1/n(n-1)AF 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 |DF|1/n(n-1)AF. 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 AF, DF and related quantities.

Original languageEnglish (US)
Pages (from-to)15-62
Number of pages48
JournalRocky Mountain Journal of Mathematics
Issue number1
StatePublished - 2000


  • Beta function
  • Diophantine inequality
  • Harmonic radius
  • Isoperimetricin equalities
  • Lagrangian plane
  • Schwarz-Christoffel transformations

ASJC Scopus subject areas

  • Mathematics(all)


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