# General Tarski-Seidenberg Theorem

The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic 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 coordinate-free 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 $$\Bbb{R}$$.)

• What does 'coordinate-free' mean? The original theorem seems coordinate-free to me …. – LSpice Apr 23 at 2:38
• @LSpice I mean a version for abstract varieties that are not defined as subsets of a Euclidean space $\Bbb{R}^n$. – KhashF Apr 23 at 2:42

The most abstract version of the Tarski-Seidenberg 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 Bochnack-Coste-Roy 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 $$\operatorname{Spec}A$$.

• Thanks! Out of curiosity, is there a notion of a "real scheme", a ringed space which is locally isomorphic with $Sper A$? – KhashF Apr 23 at 21:33
• @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) – Denis Nardin Apr 23 at 22:17