Let $n \ge 2$ and $X = (X_1, \dots, X_n)$, and set $X’:= (X_1, \dots, X_{n-1})$.
While the Normalization Theorem explains how to simplify series towards normality, it is still awkward to use directly. The reason is the factoring out of monomials at various stages, such as after using blow-up substitutions: if we want to do an induction on $\height_n(F)$ to establish some property of $F$, we can only apply the inductive hypothesis after factoring out this monomial. So, unless this property is easily preserved under multiplication by monomials (most aren’t!), we need to extract a notion of height that decreases with every substitution, without having to factor our anything.
A first possibility to do so is to simply define the new height $h_n(F)$ of $F \in \Ps{R}{X}$ by looking at the set of pairs $\left(X^r,G\right)$, with $r \in \NN^n$ and $G \in \Ps{R}{X}$, such that $F = X^r G$, and to let $h_n(F)$ be the minimum of all $\height_n(G)$ of such pairs.
However, some of the substitutions will change monomials into series that aren’t even normal: for instance, the linear substitution $l_{2,1}$ changes the monomial $X_1$ into the non-normal series $X_1+X_2$.
What we need is to consider factorizations of $F$ into pairs that remain “acceptable” factorizations after any of our substitutions. The following is the “right” definition: for $F \in \Ps{R}{X}$, we set $$\F(F):= \set{(H,G):\ H \in \Ps{R}{X’}, G \in \Ps{R}{X} \text{ and } F = HG}.$$
Exercise
Let $(H,G) \in \F(F)$ and $s$ be one of the substitutions $t_\alpha$, $p^*_{i,q}$ or $\bl_\lambda^{i,j}$ with $\lambda \ne \infty$ if $j=n$. Show that $(sH,sG) \in \F(sF)$.
On the other hand, if $s$ is $l_{n,c}$ or one of the blow-up substitutions $\bl^{i,n}_\infty$, then the conclusion of the above exercise fails in general.
However, the Normalization Theorem shows that the former is only used on a series that has infinite order in $X_n$, while the latter is used only on series that are blow-up prepared and become normal after applying it. Thus, in the latter situation, as long as we ensure that $F$ has a factorization $(H,G)$ with $H$ normal and $G$ blow-up prepared, the series $\bl_\infty F$ will be normal.
Therefore, we define $h_n(F)$ as follows: we first set $$\ord_n(F):= \min\set{\ord_{X_n}(G):\ \text{there exists } H \in \Ps{R}{X’} \text{ such that } (H,G) \in \F(F)}$$ and $$\F_1(F):= \set{(H,G) \in \F(F):\ \ord_{X_n}(G) = \ord_n(F)} \ne \emptyset.$$
Second, we set $$\tp’_n(F):= \min\set{\tp_n(G):\ \text{there exists } H \in \Ps{R}{X’} \text{ such that } (H,G) \in \F_1(F)}$$ and $$\F_2(F):= \set{(H,G) \in \F_1(F):\ \tp_n(G) = \tp’_n(F)} \ne \emptyset.$$
Third, if $\tp’_n(F) = 0$, there exists $(H,G) \in \F_2(F)$ with $G$ Tschirnhausen prepared. In this situation, we need to not only normalize the coefficients of $G$, but also $H$. Therefore, for $(H,G) \in \F_2(F)$, we set $\pn_n(H,G):= \infty$ if $G$ is not Tschirnhausen prepared and $\pn_n(H,G):= h_{n-1}\left(H\tilde G\right)$ if $G$ is Tschirnhausen prepared, where $\tilde G$ is defined for $G$ as the definition of $\pn_n(G)$. Then we set $$\pn’_n(F):= \min\set{\pn_n(H,G):\ (H,G) \in \F_2(F)}$$ and $$\F_3(F):= \set{(H,G) \in \F_2(F):\ \pn_n(H,G) = \pn’_n(F)} \ne \emptyset.$$
The remaining definitions are now straightforward: fourth, we set $$\bp’_n(F):= \min\set{\bp_n(G):\ \text{there exists } H \in \Ps{R}{X’} \text{ such that } (H,G) \in \F_3(F)}$$ and $$\F_4(F):= \set{(H,G) \in \F_3(F):\ \bp_n(G) = \bp’_n(F)} \ne \emptyset;$$
and fifth, we set $$\rd’_n(F):= \min\set{\rd_n(G):\ \text{there exists } H \in \Ps{R}{X’} \text{ such that } (H,G) \in \F_4(F)}$$ and, finally, $$h_n(F):= \begin{cases} \left(\ord_n(F), \tp’_n(F), \pn’_n(F), \bp’_n(F), \rd’_n(F)\right) &\text{if } F \text{ is not normal}, \\ 0 &\text{otherwise.} \end{cases}$$ Adapting the proof of the Normalization Theorem in the way described above, we therefore obtain:
Normalization Corollary
Let $F \in \D_n$ be nonzero and not normal. Then $h_n(F) > 0$ and one of the following holds:
- there exist $i \in \{2, \dots, n\}$ and $c \in \RR^{i-1}$ such that $\,h_n(l_{i,c} F) < h_n(F)$;
- there exist $i \in \{2, \dots, n\}$ and $\alpha \in \D_{i-1}$ such that $\alpha(0) = 0$ and $\,h_n(t_\alpha F) < h_n(F)$;
- there exist $i \in \{1, \dots, n-1\}$ and $q \in \NN$ such that $h_n\left(p^{\ast}_{i,q} F\right) < h_n(F)$ for $\ast \in \{+,-\}$;
- there exist $1 \le i < j \le n$ such that, for $\lambda \in \RR \cup \{\infty\}$, we have $\,h_n\left(\bl^{i,j}_\lambda F\right) < h_n(F)$. $\qed$
Exercise 16
Prove the Normalization Corollary.
Finally, for later use we record the following consequences of the normalization algorithm.
Lemma
Let $F \in \D_n$ be nonzero such that $h_n(F) > 0$, and let $\tau$ be a substitution as obtained from the algorithm such that $h_n(\tau F) < h_n(F)$.
- Assume there are $i \in \{1, \dots, n\}$ and $k \in \NN$ such that $\tau \in \left\{p^{+}_{i,k}, p^{-}_{i,k}\right\}$. Then $i \lt n$ and, for $q \in \NN$, we have $\,h_n\left(\tau\left(X_i^q F\right)\right) < h_n(F)$.
- Assume there are $1 \leq i < j \leq n$ and $\lambda \in \RR \cup \{\infty\}$ such that $\tau = \bl^{i,j}_\lambda$. Then for $q \in \NN$, we have $$h_n\left(\bl^{i,j}_{\infty}\left(X_j^q F\right)\right) < h_n(F)$$ and, for $\lambda \in \RR$, $$h_n\left(\bl^{i,j}_{\lambda}\left(X_i^q F\right)\right) < h_n(F).$$
If $F$ is Tschirnhausen prepared but not prenormalized, the substitution $\tau$ must be used to lower $\pn’_n(F)$; in particular, we must have $i \le n-2$. Let $(H,G) \in \F_2(F)$; then $\left(X_i^qH,G\right) \in \F_2\left(X_i^q F\right)$ and, by the inductive hypothesis, $$h_{n-1}\left(\tau \left(X_i^q H \tilde G\right)\right) \lt h_{n-1}\left(H \tilde G\right),$$ which proves the lemma in this case.
If $F$ is prenormalized but not blow-up prepared, then $i < n$ and, for $(H,G) \in \F_3(F)$, the series $H$ is normal; hence, $X_i^qH$ is normal and $\left(X_i^q H,G\right) \in \F_3\left(X_i^q F\right)$, which implies the lemma in this case. For part 2, we also proceed by induction on $n$, assuming $n>1$ and the lemma holds for $n-1$. The algorithm implies that $F$ is blow-up prepared, so if $j \lt n$, the lemma holds by the inductive hypothesis. So we assume that $j=n$; since $\bl^{i,n}_\infty \left(X_n^q F\right) = X_n^q \cdot \bl^{i,n}_\infty F$ in this case, we obtain the first statement of the lemma. On the other hand, if $\lambda \in \RR$ then, since $i \lt n$, we have $\left(X_i^q H,G\right) \in \F_4\left(X_i^q F\right)$ for $(H,g) \in \F_4(F)$, so the lemma also follows in this case. $\qed$