Tschirnhausen transformation
A first measure describing how far from normal a series $F \in \Ps{R}{X,Y}$ in two indeterminates $X$ and $Y$ is is given by $$\ord_Y(F):= \ord(F(0,Y)).$$ Exercise Show that $\ord_Y(F) = 0$ if and only if $F$ is a unit, while, if $\ord_Y(F) = d>0$, there exists a unit $U \in \Ps{R}{X,Y}$ such that $$F =…