Suppose that I have $N$ a bidimensional analytical manifold, $\mathcal{F}$ a foliation in $N$, and let $P\in N$. Being $\mathcal{O}$ the local ring of germs of holomorphic functions in $P$, and $\mathcal{M}\subset\mathcal{O}$ its maximal ideal.
For $f\in\mathcal{O}$, we define the order of $f$ at P as $$ v_P(f)=max\{t;f\in\mathcal{M}^t\} $$ which is also understood as the multiplicity of the zero of f at $P$.
I dont know what $\mathcal{M}^t$ means, though I know that, if I write $f=\sum f_{ij}x^ix^j$, we have $$ v_P(f)=min\{i+j;f_{ij}\}\neq 0 $$ Can somebody please explain to me what does $\mathcal{M}^t$ means?
