Associated forms and hypersurface singularities: The binary case

In the recent articles [1, 5], it was conjectured that all rational GLn-invariant functions of forms of degree d ≥ 3 on ℂⁿ can be extracted, in a canonical way, from those of forms of degree n(d - 2) by means of assigning to every form with nonvanishing discriminant the so-called associated form. The conjecture was confirmed in [5] for binary forms of degree d ≤ 6 as well as for ternary cubics. Furthermore, a weaker version of it was settled in [1] for arbitrary n and d. In the present paper, we focus on the case n = 2 and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree.