Normalization: the general case

Based on our discussion here, we can now adapt the two-variable normalization to more than two variables as follows.

Assume $n > 2$, and let $F \in \Ps{R}{X}$ and $d \in \NN$.

  1. We call $F$ Tschirnhausen prepared of order $d$ in $X_n$ if, written as a series in $\Ps{R}{X’}[\![X_n]\!]$, we have $$F = \sum_{k=0}^{d-2} F_k(X’) X_n^k + X_n^d U,$$ where $U \in \Ps{R}{X}$ is a unit and $F_k(0) = 0$ for $k=0, \dots, d-2$.
  2. We call $F$ prenormalized of order $d$ in $X_n$ if $F$ is Tschirnhausen prepared of order $d$ in $X_n$ such that the set $\{F_0, \dots, F_{d-2}\}$ is normal.
  3. We call $F$ blow-up prepared in $X_i$ of order $d$ in $X_n$ if $F$ is prenormalized of order $d$ in $X_n$ and, for $k=0, \dots, d-2$, the series $F_k$ is divisible by $X_i^{d-k}$.

Let now $F = \sum_{k=0}^\infty F_k(X’) X_n^k \in \Ps{R}{X}$ and assume that $\height_{n-1}:\Ps{R}{X’} \into \left(\NN \cup \{\infty\}\right)^{4(n-1)}$ has been defined. We define

  • $\tp_n(F):= 0$ if $F$ is Tschirnhausen prepared, and $\tp(F):= 1$ otherwise;
  • $\pn_n(F):= (\infty, \dots, \infty)$ if $F$ is not Tschirnhausen prepared, and $\pn_n(F):= \height_{n-1}\left(\tilde F\right)$ if $F$ is Tschirnhausen prepared of order $d$ in $X_n$ and $\tilde F$ is the product of all $F_i-F_j$, for $1 \le i < j \le d-1$;
  • $\bp_n(F):= \infty$ if $F$ is not prenormalized, and $$\bp(F):= \left|\set{i = 1, \dots, n-1:\ F \text{ is not blow-up prepared in } X_i} \right|$$ otherwise;
  • $\rd_n(F):= \infty$ if $F$ is not blow-up prepared, and if $F$ is blow-up prepared of order $d < \infty$ in $X_n$, we set $K:= \set{k = 0, \dots, d-2:\ F_k \ne 0}$ and, for $k \in K$, we write $F_k = \left(X'\right)^{r_k} U_k$ with $r_k \in \NN^{n-1}$ and $U_k \in \Ps{R}{X'}$ a unit, and we set $$\rd(F):= \min\left(\set{\frac{\sum r_k}{d-k}:\ k \in K} \cup \{0\}\right),$$ where $\sum r_k:= r_{k,1} + \cdots + r_{k,n-1}$.

Finally, we set $$\height_n(F):= \left(\ord_{X_n}(F), \tp_n(F), \pn_n(F), \bp_n(F), \rd_n(F)\right),$$ considered as an element of $(\NN \cup \{\infty\})^{4n}$ equiped with its lexicographic ordering.

As in this remark, we now fix subrings $\D_n \subseteq \Ps{R}{X}$, for each $n \in \NN$, such that


$\RR[X] \subseteq \D_n \subseteq \D_{n+1}$ for all $n$;


if $F \in \D_m$ and $G_1, \dots, G_m \in \D_n$ such that $G_1(0) = \cdots = G_m(0) = 0$, then $F(G_1, \dots, G_m) \in \D_n$;


each $\D_n$ is closed under differentiation;


if $F \in \D_n$ and $G \in \Ps{R}{X}$ are such that $F = X_i \cdot G$ for some $i \leq n$, then $G \in \D_n$;


if $n>0$ and $F \in \D_n$ are such that $F(0) = 0$ and $(\partial F/\partial X_n) (0) \neq 0$, there is an $\alpha \in \D_{n-1}$ with $\alpha(0) = 0$ such that $F\left(X’,\alpha\left(X’\right)\right) = 0$.

Taking into account the changes to the two-variable normalization outlined here, we can now prove the following:

Normalization Theorem
Let $F \in \D_n$ be nonzero and not normal. Then $\height_n(f) > 0$ and one of the following holds:

  1. there exist $i \in \{2, \dots, n\}$ and $c \in \RR^{i-1}$ such that $\,\height_n(l_{i,c} F) < \height_n(F)$;
  2. there exist $i \in \{2, \dots, n\}$ and $\alpha \in \D_{i-1}$ such that $\alpha(0) = 0$ and $\,\height_n(t_\alpha F) < \height_n(F)$;
  3. there exist $i \in \{1, \dots, n-1\}$ and $q \in \NN$ such that $\height_n\left(p^{\ast}_{i,q} F\right) < \height_n(F)$ for $\ast \in \{+,-\}$;
  4. there exist $1 \le i < j < n$ such that, for $\lambda \in \RR \cup \{\infty\}$, there exist $r = r(\lambda) \in \NN^{n-1}$ and $G = G(\lambda) \in \D_n$ such that $\,\height_n(G) < \height_n(F)$ and $\,\bl^{i,j}_\lambda F = (X')^r G$;
  5. there exists $1 \le i < n$ such that $\,\bl^{i,n}_\infty F$ is normal and, for $\lambda \in \RR$, there exist $r = r(\lambda) \in \NN^{n-1}$ and $G = G(\lambda) \in \D_n$ such that $\,\height_n(G) < \height_n(F)$ and $\,\bl^{i,n}_\lambda F = (X')^r G$.
(Show proof)
We write $F = \sum_{k=0}^\infty F_k(X’) X_n^k$ and set $d:= \ord_{X_n}(F)$, and we proceed by induction on $n$; the case $n=2$ follows from this proposition and remark, so we assume $n>2$ and the theorem holds for $n-1$. We distinguish five cases:

Case 1: $d = \infty$. Then Statement 1 holds by this lemma.

Case 2: $d < \infty$, but $F$ is not Tschirnhausen prepared. Then Statement 2 holds by an easy adaptation of the argument used to prove Lemma 2.

Case 3: $d < \infty$ and $F$ is Tschirnhausen prepared, but not prenormalized. Then by the inductive hypothesis and the definition of $\height_n$, one of the four statements holds. Case 4: $d < \infty$ and $F$ is prenormalized, but not blow-up prepared. Then Statement 3 holds with $q:= d!$, for some $i \in \{1, \dots, n-1\}$. Case 5: $d < \infty$ and $F$ is blow-up prepared. In this case, we proceed in analogy with the proof of Case 4 of this proposition to show that Statement 4 holds, choosing any $i \in \{1, \dots, n-1\}$ that witnesses $\rd_n(F) > 0$ and taking $j := n$. $\qed$

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