Inequalities can be turned into equations - semi-algebraic varieties turned into $\mathbb{R}$-schemes?

Question

Suppose that a semi-algebraic set in $X \subset \mathbb{R}^n$ is defined by some polynomial equalities and inequalities, such as $p(x) \geq q(x)$. We can reduce these to inequalities of the form $h(x) \geq 0$, and then we can think of this as the statement that $h(x) = t^2$, where $t$ is now a new variable. Hence, we get a scheme over $\mathbb{R}$, whose real points naturally correspond to X (by forgetting the $t$-coordinates). Denote by $X^s$ this scheme.

My question: To what extent is the map $X \to X^s$ an embedding of categories? (A morphism between semi-algebraic sets is something whose graph is algebraic? I'm not really sure what it should be.)

Answer 1

Your argument shows (with some complements in order to deal with inequations and strict inequalities) that every semialgebraic set $X$ is the projection of some real algebraic set $V$. But there is no canonical such algebraic set as you seem to believe. It depends on the description of the semialgebraic set $X$, and so there is no such map $X \mapsto X^s$.

There is a way to associate canonically a scheme to a semialgebraic set $X$: one can consider the affine scheme of the ring of semialgebraic continuous functions on $X$. It works pretty well for locally closed semialgebraic sets, but such a scheme is far away from usual $\mathbb{R}$-schemes in algebraic geometry. On this topic you can see the work by Niels Schwartz, he has written a Memoir of the AMS on real closed spaces.

Schwartz, Niels. The basic theory of real closed spaces. Mem. Amer. Math. Soc. 77 (1989), no. 397, viii+122 pp.

Do you think you could give an example where the $X^s$ are different (as isomorphism classes of schemes, not as an embedding)? (For me this has the feeling of the trick for embedding complements of affine hypersurfaces as closed subvarieties by bumping the dimension up by one.)

If $X$ is the red closed semialgebraic set hereunder, what is $X^s$?