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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 SL2${\rm SL}_2$‐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 SL2(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.
Bulletin of the London Mathematical Society – Wiley
Published: Aug 1, 2022
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