## Abstract

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 GL_{2}(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 GL_{2}(C) invarianc ε of |D_{F}|^{1/n(n-1)}A_{F} with respect to Lagrangian planes in C^{2} (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.

Original language | English (US) |
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Pages (from-to) | 15-62 |

Number of pages | 48 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)