Computing the adjunction divisor

For any (closed) singular place we determine the multiplicity $d_{[Q]}:=ord_Q ({\mathcal{A}})$, alternatively, by the formula (2) or the Dedekind formula (cf. Remarks 2.5 and 2.3). Note that to compute a local intersection number $i_P(f,g)$ (as appearing in both formulas), we can proceed inductively, computing the primitive parametrization $\varphi:K[[x,y]]\to K[[t]]$ of $g$ up to degree $k$, until $ord_t(\varphi(x){\:\text{mod}\:} t^k,\varphi(y){\:\text{mod}\:} t^k)$ is less than $k/ord(f)$.