Associated forms and hypersurface singularities: The binary case

Abstract

In the recent articles [5, 1], 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.