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

In 1848, Hermite introduced a reduction theory for binary forms of degree (Formula presented.) 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 (Formula presented.) -invariant of binary (Formula presented.) -ic forms defined over (Formula presented.), which is now known as the Julia invariant. In this paper, for each (Formula presented.) and (Formula presented.) with (Formula presented.), we determine the asymptotic behavior of the number of (Formula presented.) -equivalence classes of binary (Formula presented.) -ic forms, with (Formula presented.) pairs of complex roots, having bounded Julia invariant. Specializing to (Formula presented.) 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.