Associated forms and hypersurface singularities: The binary case

Abstract

Abstract In the recent articles [5, 1], it was conjectured that all rational {\mathrm{GL}_{n}} -invariant functions of forms of degree {d\geq 3} on {\mathbb{C}^{n}} 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\leq 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.