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Semialgebraic Set

A semialgebraic set is a subset of R^n which is a finite Boolean combination of sets of the form {x:f(x)>0} and {x:g(x)=0}, where f and g are polynomials over the real numbers.

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

{x in R^n:f_i(x)=0 (i=1,...,r), g_j(x)>0 (j=1,...,s)}.

A basic closed semialgebraic set has a description

{x in R^n:g_j(x)>=0 (j=1,...,s)}.

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.

See also

Cylindrical Algebraic Decomposition, Polynomial Optimization, Quantifier Elimination, Tarski's Theorem

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References

Basu, S.; Pollack, R.; and Roy, M.-F. Algorithms in Real Algebraic Geometry, 2nd ed. Berlin, Germany: Springer-Verlag, 2006.Bierstone, E. and Milman, P. "Semialgebraic and Subanalytic Sets." IHES Pub. Math. 67, 5-42, 1988.Marker, D. "Model Theory and Exponentiation." Not. Amer. Math. Soc. 43, 753-759, 1996.Strzebonski, A. "Solving Algebraic Inequalities." Mathematica J. 7, 525-541, 2000.

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Semialgebraic Set

Cite this as:

Weisstein, Eric W. "Semialgebraic Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SemialgebraicSet.html

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