a large number of components. Moreover, even in the presence of a general solution for this non-linear algebraic system, the problem of phase coexistence would still remain out of reach in view of the previous remarks.

In a series of recent papers [Gazzillo00,Gazzillo02,Gazzillo03,Fantoni05,Fantoni06,Gazzillo06]{}, we have attempted to make some progress along two different lines.

First, starting from the *density* expansion of the cavity function at contact, we have considered a sequence of simpler approximations (compared to the PY one) [Gazzillo00,Gazzillo02,Gazzillo03,Gazzillo04,Fantoni05]{}. Within the two simplest ones among these approximations, denoted as $C_{0}$ and $C_{1}$ (for reasons which will become clear in the following), we have been able to derive analytically all relevant information regarding structure and thermodynamics, including the phase coexistence, in view of the fact that the corresponding free energy turns out to be truncatable [@Fantoni05]. Due to the simplicity of $C_{0}$ and $C_{1}$, it is however reasonable to expect these approximations to fail at high packing fractions, with a consequently incomplete or even incorrect description of the effects of polydispersity on the phase diagram.

Therefore, in collaboration with Peter Sollich, we have recently explored a second approach [@Fantoni06], where the expansion variable (which must be small) is an appropriate 