## Normalization in two variables

We are now ready to give a measure of “distance from normality” for series in $\Ps{R}{X,Y}$. Definition We call $F \in \Ps{R}{X,Y}$ blow-up prepared if $F$ is Tschirnhausen prepared of order $d$ in $Y$ and, for $k=0, \dots, d-2$, the series $\left(\partial^k F/\partial Y^k\right)(X,0)$ is divisible by $X^{d-k}$. Let now \$F = \sum_{k=0}^\infty F_k(X) Y^k…