Existential theory of the reals

In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form ∃ X 1 ⋯ ∃ X n F ( X 1 , … , X n ) , {\displaystyle \exists X_{1}\cdots \exists X_{n}\,F(X_{1},\dots ,X_{n}),} where the variables X i {\displaystyle X_{i}} are interpreted as having real number values, and where F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F {\displaystyle F} , make it become true.

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