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For a smooth, irreducible projective surface S over \mathbb{C}, the number ofr-nodal curves in an ample linear system |L| (where L is a line bundle on S)can be expressed using the rth Bell polynomial P_{r} in r universal functionsa_{i} of (S,L), which are linear polynomials in the four Chern numbers of S andL. We use this result to establish a proof of the classical shape conjecturesof Di Francesco-Itzykson and G\"ottsche governing node polynomials in the caseof P^{2}. We also give a recursive procedure which provides the L^{2}-term ofthe polynomials a_{i}.

The Di Francesco-Itzykson-G\"ottsche Conjectures for Node Polynomials of $\mathbb{P}^{2}$