Patrick Speissegger
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 $$F =…
In this post, we introduce a type of substitutions to be used in our normalization algorithm: let $X$ and $Y$ be two single indeterminates. Definition For $\lambda \in \RR$, we let $\bl_\lambda:\Ps{R}{X,Y} \into \Ps{R}{X,Y}$ be the blow-up substitution defined by $$\bl_\lambda(X):= X \quad\text{and}\quad \bl_\lambda(Y):= X(\lambda+Y).$$ We also let $\bl_\infty:\Ps{R}{X,Y} \into \Ps{R}{X,Y}$ be the blow-up substitution…
The first step towards proving the o-minimality of $\Ran$ is to show that the quantifier-free definable sets have finitely many connected components. As discussed in this post, this means (essentially) that we need to show that basic $\Pc{R}{X}$-sets have finitely many connected components. Example 1 Let $\alpha \in \NN^n$ and $r \in (0,\infty)^n$. Show that…
Let $r \in (0,\infty)^n$ be a polyradius. For $F \in \Pc{R}{X}_r$, we denote by $F_r:\bar B(r) \into \RR$ the function defined by $F_r(x):= F(x)$. We denote by $C^\infty(r)$ the ring of all $C^\infty$ functions on $B(r)$. For $f \in C^\infty(r)$, we let $T_r(f) \in \Ps{R}{X}$ be the Taylor expansion of $f$ at $0$; note that…
Our next goal is to give a different construction method for o-minimal structures. The main example discussed for this construction method is the expansion $\Ran$ of the real field by all restricted analytic functions, although we will mention some other expansions to which this same method applies. To achieve this, we will again work in…