Patrick Speissegger
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 $$F = \sum_{k=0}^{d-2}…
We now discuss normalization in any number of variables, in the axiomatic setting suggested by this corollary. From now on, we set $X = (X_1, \dots, X_n)$ and $X’:= (X_1, \dots, X_{n-1})$. One of the key ingredients to do this is that normalization of one series implies normalization of finitely many series: Definition Let $F_1,…
…and very brief it is—since I only started this blog in January 2015. It arose out of my attempt to finally make good on my promise (to nobody in particular, really) to write a sequel to Lou’s book. The idea of this sequel is to present the main ideas used in the construction of o-minimal…
We are now ready to give a measure of “distance from normality” for series in $\Ps{R}{X,Y}$. Definition We call $F \in \Ps{R}{X,Y}$ blow-up prepared if $F$ is Tschirnhausen prepared of order $d$ in $Y$ and, for $k=0, \dots, d-2$, the series $\left(\partial^k F/\partial Y^k\right)(X,0)$ is divisible by $X^{d-k}$. Let now $F = \sum_{k=0}^\infty F_k(X) Y^k…
A first measure describing how far from normal a series $F \in \Ps{R}{X,Y}$ in two indeterminates $X$ and $Y$ is is given by $$\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 =…