On the number of integral binary n-ic forms having bounded Julia invariant

2013/12/27 by Manjul Bhargava, Andrew Yang, Bhargava, Manjul +1
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1312.7339

Abstract

In 1848, Hermite introduced a reduction theory for binary forms of degree n which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational SL₂-invariant of binary n-ic forms defined over ℝ, which is now known as the Julia invariant. In this paper, for each n and k with n+k≥ 3, we determine the asymptotic behavior of the number of SL₂(ℤ)-equivalence classes of binary n-ic forms, with k pairs of complex roots, having bounded Julia invariant. Specializing to (n,k)=(2,1) and (3,0), respectively, recovers the asymptotic results of Gauss and Davenport on positive definite binary quadratic forms and positive discriminant binary cubic forms, respectively.

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