The existential theory of the reals is the decision problem of whether a sentence 
is true, where 
is a Boolean combination of polynomial
equalities and inequalities. Integer coefficients
are encoded finitely, so the question defines a discrete decision
problem. The class 
consists of problems polynomial-time
many-one reducible to this truth problem. For example, 
is false, whereas 
is true. Canny (1988) proved decidability
in polynomial space. Schaefer et al. (2024)
survey the resulting complexity class.
Existential Theory of the Reals
See also
Arithmetic Circuit, Counting Hierarchy, Decision Problem, Quantifier EliminationExplore with Wolfram|Alpha
References
Canny, J. F. "Some Algebraic and Geometric Computations in PSPACE." Technical Report UCB/CSD-88-439. Berkeley, CA: University of California, 1988. https://www2.eecs.berkeley.edu/Pubs/TechRpts/1988/6041.html.Schaefer, M.; Cardinal, J.; and Miltzow, T. "The Existential Theory of the Reals as a Complexity Class: A Compendium." 25 Jul 2024. https://arxiv.org/abs/2407.18006.Cite this as:
Weisstein, Eric W. "Existential Theory of the Reals." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExistentialTheoryoftheReals.html