Tarski–Seidenberg theorem

Remarks

The theorem is the geometric form of quantifier elimination for real closed fields. The set $\pi(S)$ is defined by the formula $\exists x_{n+1} \, \varphi(x_1, \dots, x_{n+1})$, where $\varphi$ is a quantifier-free formula that defines $S$. Quantifier elimination replaces it by a quantifier-free formula with parameters in $R$, and the set defined by such a formula is semialgebraic.

Example: let $S = \set{(a, x) \in \R^2 : x^2 - a = 0}$, which is semialgebraic. Its projection to the first coordinate is $\set{a : a \ge 0}$, which is semialgebraic as the union of $\set{a : a = 0}$ and $\set{a : a > 0}$.

The analogous statement fails for algebraic sets. The projection of the algebraic set $S$ above is $[0, \infty)$, which is not the zero set of any family of polynomials in one variable. This is why inequalities are part of the definition.

Going deeper

For a student

By induction on $n$, the image of a semialgebraic set under any polynomial map is semialgebraic, and so are closures, interiors and sets defined with quantifiers over $R$. This makes semialgebraic sets the natural setting of real algebraic geometry, in contrast with real algebraic sets, which lack this closure property. It also shows that every subset of $R$ definable with $+$, $\cdot$ and $<$ is semialgebraic, hence a finite union of points and open intervals. This is the starting point of o-minimality. Algorithms for quantifier elimination over $\R$, such as cylindrical algebraic decomposition, make the projection effective.