  In the recent articles by Alper, Eastwood and Isaev, it was conjectured that
all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge
3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms
of degree $n(d-2)$ by means of assigning every form with nonvanishing
discriminant the so-called associated form. While this surprising statement is
interesting from the point of view of classical invariant theory, its original
motivation was the reconstruction problem for isolated hypersurface
singularities, which is the problem of finding a constructive proof of the
well-known Mather-Yau theorem. The conjecture was confirmed by Eastwood and
Isaev for binary forms of degree $d \le 6$ as well as ternary cubics.
Furthermore, a weaker version of it was settled by Alper and Isaev 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. This result allows one to extract a complete system of
biholomorphic invariants of homogeneous plane curve singularities from their
Milnor algebras.
