COMMENT ⊗   VALID 00002 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\input basic
C00004 ENDMK
C⊗;
\input basic
\chcode'27=13 \def↔{\penalty999\ }
\def\vmake (#1,#2){(#1,\ldots,#2)}
\def\affsp #1{\hbox{\bf A}↑#1}
\def\Rf{\hbox{\bf R}}
\def\veclet (#1){\hbox{$\overline{\hbox{\bf #1}}$}}
\def\Xspace{\hbox{\bf X}}

\ctrline{\bf EMBEDDING SEMI-ALGEBRAIC SPACES}
\vskip 12pt
 
Let \Rf\ be a real closed field and $\affsp n$ be $n$-dimensional
affine \Rf-space. Define $\rho\vmake(x↓1,x↓n)$ to be 
$\sqrt{x↑2↓1+\ldots+x↑2↓n}$. If $f↓i,g↓i \in \Rf[x↓1,\ldots,x↓n],\ 
\Zscr \{ f↓i \} = \{ \veclet(x)\in \affsp n | f↓i(\veclet(x))=0
\hbox{, all } f↓i \} \hbox{ and } \Uscr \{ g↓i \}=\{ \veclet(x)\in \affsp n
 g↓i(\veclet(x))>0 \hbox{, all } g↓i \}$.

\vfill\end