Semialgebraic Set

A semialgebraic set is a subset of which is a finite Boolean combination of sets of the form and , where and are polynomials over the real numbers.

A basic semialgebraic set is defined by a finite conjunction of polynomial equalities and inequalities, for example

A basic closed semialgebraic set has a description

Every semialgebraic set is a finite union of basic semialgebraic sets (Basu et al. 2006).

By Tarski's theorem, the solution set of a quantified system of algebraic equations and inequalities over the real numbers is a semialgebraic set (Strzebonski 2000). In particular, the feasible region of a polynomial optimization problem is a basic semialgebraic set.

Semialgebraic sets are closed under coordinate projections. Indeed, membership in the projection of can be expressed as the existential quantifier condition that there is a with , and Tarski's theorem removes that quantifier. For example, the projection of the unit circle onto the -axis is the closed interval .

In one dimension, semialgebraic sets are exactly the finite unions of points and intervals, which may be open, closed, half-open, or unbounded (Plaumann 2023, p. 5). In higher dimensions, cylindrical algebraic decomposition provides a finite description by cells. A semialgebraic function is characterized by its function graph being a semialgebraic set.