On the number of integral binary nn‐ic forms having bounded Julia invariant. Issue 4 (17th May 2022) - Bhargava, Manjul; Yang, Andrew
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 SL_2 ‐invariant of binary n n ‐ic forms defined over R R, which is now known as the Julia invariant. In this paper, for each n n and k k with n + k ⩾ 3 n+k3, we determine the asymptotic behavior of the number of SL 2 ( Z ) SL_2(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.