Patrick Speissegger
Finally, we get to Step 3 of the proof of this theorem: establishing a theorem of the complement for the “right” collection of existentially definable sets. We start with a few exercises. Exercises Let $\tau_n:\RR^n \into (-1,1)^n$ be the semialgebraic diffeomorphism defined here. Show that, for every semialgebraic set $A \subseteq \RR^n$, the set $\tau_n(A)$…
We now want to obtain a description of the projections of bounded semi-$\C$-sets. By this corollary, we only need consider projections of trivial semi-$\C$-sets, that is, of trivial $\C$-sets. We start with a few remarks about $\C$-manifolds: let $M \subseteq \RR^n$ be a nonempty $\C$-manifold of dimension $m$. First, let $r$ be a polyradius associated…
We continue working with our rings of germs $\C_n$, for $n \in \NN$, as introduced here. As with convergent power series, for $f \in \C_n$ such that the polyradius $(1, \dots 1)$ is $f$-admissible, we define a function $\bar f:\RR^n \into \RR$ by setting $$\bar f(x):= \begin{cases} f(x) &\text{if } x \in \bar B(1), \\…
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…