Real Semialgebraic Sets

Definition

A set $A \in \R ^n$ is semialgebraic if there exist polynomials $P_1, \dots , P_n$ and polynomials $Q_1, \dots , Q_m$ such that

\[ A = \bigcup_{i = 1}^{n} \Set{x \in \R ^n}{P_i(x) = 0} \cup \bigcup_{j = 1}^{m} \Set{x \in \R ^{n}}{Q_j(x) \leq 0} \]

In other words, a semialgebraic set can be written as a finite union of sets defined by polynomial equalities and inequalities.