A set $A \in \R ^n$ is algebraic if there exist polynomials $P_1, \dots , P_n$ such that \[ A = \bigcup_{i = 1}^{n} \Set{x \in \R ^n}{P_i(x) = 0} \] In other words, a semialgebraic set can be written as a finite union of sets defined by polynomial equations.
\[ A = \bigcup_{i = 1}^{n} \Set{x \in \R ^n}{P_i(x) = 0} \]