  \[thm:case0\]Assume that the map from Theorem \[thm:injection\] is injective. We then have the lower bound $$\begin{aligned}     \sum_{N>0}\dim_{\mathbb Q}\ker C_{N,0}\cdot x^N\geq 0 \mathbb O(x)^0\mathbb S(x)-\mathbb S(x)^0{\,}.\end{aligned}$$ 