be expressed uniquely as a polynomial of degree $n$ in the variables $q_{a,b}$. Choose some normal ordering of the corresponding polynomial in the variables $\Omega_{a,b}$, and call this vertex operator $-D^{2n}_{I,J}$. Then $D^{2n+2}_{I,J} + D^{2n}_{I,J}$ has the property that $\pi_{n}(D^{2n+2}_{I,J} + D^{2n}_{I,J})\in (\cH(n)^{O(n)})_{(2n-2)}.$ Continuing this process, we arrive at a vertex operator $\sum_{k=1}^{n+1} D^{2k}_{I,J}$ in the kernel of $\pi_{n}$. We must have $$\label{decompofd} D_{I,J} = \sum_{k=1}^{n+1}D^{2k}_{I,J},$$ since $D_{I,J}$ is uniquely characterized by (\[uniquedij\]).

In this decomposition, the term $D^2_{I,J}$ lies in the space $A_m$ spanned by $\{\Omega_{a,b}|~a+b=m\}$, for $m = |I| +|J|+2n$. Recall that for $m$ even, $A_m  = \partial^2 A_{m-2} \oplus \bra J^{m}\ket$, and for $m$ odd, $A_m  = \partial^3 A_{m-3} \oplus \bra \partial J^{m-1}\ket$. For $m$ even (respectively odd), define $pr_m: A_m\ra \bra J^m\ket$ (respectively $pr_m: A_m\ra \bra \partial J^{m-1}\ket$) to be the projection onto the second term. Define the [*remainder*]{} $$\label{defofrij} R_{I,J} = pr_m(D^2_{I,J}).$$

\[uniquenessofr\] Given $D_{I,J}\in\cI_n$ as above, suppose that $D_{I,J} = \sum_{k=1}^{n+1} D^{2k}_{I,J}$ and $D_{I,J} = \sum_{k=1}^{n+1} \tilde{D}^{2k}_{I,J}$ are two different decompositions of $D_{I,J}$ of the form (\[decompofd\]). Then $$D^2_{I,J} - \tilde{D}^2_{I,J} \in \partial^2 (A_{m-2}),$$ where $m = |I| +|J| +2 n$. In particular, $R_{I,J}$ is independent of the choice of decomposition of $D_{I,J}$.

This 