Patrick Speissegger
We now return to $\Ps{R}{X}$-sets or, more generally, to so-called $\C$-sets, where $\C = (\C_n)_{n \in \NN}$ is a collection of subrings $\C_n$ of $C^\infty_n$ obtained as follows: for every polyradius $r \in (0,\infty)^n$, we assume being given a subring $\C_r$ of $C^\infty_r$ such that, (C1) for $f \in \C_r$, there exists $s > r$…
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…
Based on our discussion here, we can now adapt the two-variable normalization to more than two variables as follows. Definition Assume $n > 2$, and let $F \in \Ps{R}{X}$ and $d \in \NN$. 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}…
We now discuss normalization in any number of variables, in the axiomatic setting suggested by this corollary. From now on, we set $X = (X_1, \dots, X_n)$ and $X’:= (X_1, \dots, X_{n-1})$. One of the key ingredients to do this is that normalization of one series implies normalization of finitely many series: Definition Let $F_1,…
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…