There is a theorem stated as, Let k be a real closed field and D subset of k be a subring of k. Given a semi-algebraic set of kn+1 defined over D, its projection to kn is a semi-algebraic set defined over D. Can some one please explain what does this mean, and if possible how to visualize this geometrically
The theorem in question is due to Tarski-Seidenberg. Suppose you have a finite disjunction of conjunctions of polynomial equalities and inequalities in $k^{n+1}$
$$ S(x_1,\ldots, x_n, x_{n+1}) \. $$
The projection to $k^n$ would be given by
$$ \exists x_{n+1}: S(x1,\ldots, x_{n+1}) \; .$$
As Peter Breuer comments, this would correspond to an infinite disjunction of algebraic formulas. However, it is possible to "eliminate the existence quantor", i.e. to find an equivalent finite disjunction of formulas. For the proof of this elimination one needs an algebraic expression of the coefficients of a polynomial of one variable (the variable which vanishes by projection) that allows to decide whether the polynomial has a zero. A sturm chain provides such a device and is used in the proof you'll find e.g. in
https://perso.univ-rennes1.fr/michel.coste/polyens/SAG.pdf
The calculations are algorithmic, I can't conceive of an easy visualisation. But it might help to have a look at Coste's examples.
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