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

Abstract

In 1848, Hermite introduced a reduction theory for binary forms of degree n $n$ 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 SL 2 ${\rm SL}_2$ -invariant of binary n $n$ -ic forms defined over R $\mathbb {R}$ , which is now known as the Julia invariant. In this paper, for each n $n$ and k $k$ with n + k ⩾ 3 $n+k\geqslant 3$ , we determine the asymptotic behavior of the number of SL 2 ( Z ) ${\rm SL}_2(\mathbb {Z})$ -equivalence classes of binary n $n$ -ic forms, with k $k$ pairs of complex roots, having bounded Julia invariant. Specializing to ( n , k ) = ( 2 , 1 ) $(n,k)=(2,1)$ 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.