A reconstruction of Euler data

We apply the mirror principle (see Mirror principle, I, Asian J. Math. 1 (1997), pp. 729–763) to reconstruct the Euler data $Q=\{Q_d\}_{d\in {\mathbb N}\cup \{0\}}$ associated to a vector bundle $V$ on ${\mathbb C}{\mathrm P}^n$ and a multiplicative class $b$. This gives a direct way to compute the intersection number $K_d$ without referring to any other Euler data linked to $Q$. Here $K_d$ is the integral of the cohomology class $b(V_d)$ of the induced bundle $V_d$ on a stable map moduli space. A package “EulerData_MP.m” in Maple V that carries out the actual computation is provided in the electronic version math.AG/0003071 of the current paper. For $b$, the Chern polynomial, the computation of $K_1$ for the bundle $V=T_{\ast }{\mathbb C}{\mathrm P}^2$, and $K_d$, $d=1,2,3$, for the bundles ${\mathcal O}_{{\mathbb C}{\mathrm P}^4}(l)$ with $6\le l\le 10$ are done using the code and are also included.