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

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 ${\mathrm{SL}}_2$-invariant of binary $n$-ic forms defined over $\mathbb{R}$, which is now known as the Julia invariant. In this paper, for each $n$ and $k$ with $n+k\ge 3$, we determine the asymptotic behavior of the number of ${\mathrm{SL}}_2(\mathbb{Z})$-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.

17 pages; to appear in Bull. Lond. Math. Soc

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