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

On the number of integral binary n$n$‐ic forms having bounded Julia invariant 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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

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References (36)

Publisher
Wiley
Copyright
© 2022 London Mathematical Society.
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12625
Publisher site
See Article on Publisher Site

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 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.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Aug 1, 2022

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