The Di Francesco–itzykson–göttsche Conjectures for Node Polynomials of ℙ²

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system [Formula: see text] (where [Formula: see text] is a line bundle on S) can be expressed using the rth Bell polynomial P r in universal functions a i , 1 ≤ i ≤ r, of (S, [Formula: see text]), which are ℤ-linear polynomials in the four Chern numbers of S and [Formula: see text]. We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ 2 . We also give a recursive procedure which provides the [Formula: see text]-term of the polynomials a i .