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Semialgebraic set

A subset of real affine space described by finitely many polynomial equalities and inequalities.

A subset S⊆RnS\subseteq\mathbb R^n is semialgebraic if it can be obtained from finitely many sets of the form

{x:p(x)=0}and{x:q(x)>0},\{x:p(x)=0\}\quad\text{and}\quad\{x:q(x)>0\},

where p,q∈R[x1,…,xn]p,q\in\mathbb R[x_1,\ldots,x_n], using finitely many unions, intersections, and complements. Equivalently, membership in SS is given by a finite Boolean combination of polynomial sign conditions.

Semialgebraic sets are closed under products, projections, closure, and taking connected components. They form the basic class of sets in real algebraic geometry.