The TarskiSeidenberg Theorem states that the polynomial image of a semialgebraic set is semialgebraic. A semialgebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of subsets of the form $$ \{P_1=\dots=P_k=0, Q_1>0,\dots,Q_l>0\} $$ where $P_i$'s and $Q_j$'s belong to $\Bbb{R}[x_1,\dots,x_n]$. I wonder is there a general coordinatefree version of this theorem for morphisms of real varieties? (By a real variety I mean the set of $\Bbb{R}$points of a variety over $\Bbb{R}$.)

2$\begingroup$ What does 'coordinatefree' mean? The original theorem seems coordinatefree to me …. $\endgroup$ – LSpice Apr 23 at 2:38

1$\begingroup$ @LSpice I mean a version for abstract varieties that are not defined as subsets of a Euclidean space $\Bbb{R}^n$. $\endgroup$ – KhashF Apr 23 at 2:42
The most abstract version of the TarskiSeidenberg theorem I know of is the following
Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map
$$f^*:\operatorname{Sper}B\to \operatorname{Sper}A$$
sends constructible sets to constructible sets.
Here $\operatorname{Sper}$ is the real spectrum of the ring, i.e. the set of all pairs $(p,<)$ where $p$ is a prime ideal and $<$ is an order on the residue field at $p$.
It is well known (e.g. theorem 7.2.3 in BochnackCosteRoy Real Algebraic Geometry) that if $A$ is an algebra of finite presentation over $\mathbb{R}$, the boolean algebra of constructible subsets of $\operatorname{Sper}A$ is in natural bijection with the semialgebraic subsets of the real points of the variety $\operatorname{Spec}A$.

$\begingroup$ Thanks! Out of curiosity, is there a notion of a "real scheme", a ringed space which is locally isomorphic with $Sper A$? $\endgroup$ – KhashF Apr 23 at 21:33

1$\begingroup$ @KhashF Yes, they are called real closed spaces and they have been developed by Niels Schwartz (they are not really locally modeled on $\operatorname{Sper}A$, rather on proconstructible subsets of it, but the idea is the same) $\endgroup$ – Denis Nardin Apr 23 at 22:17