We introduce the complexity class ∃R based on the existential theory of the reals. We show that the definition of ∃R is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known problems turn out to be complete for ∃R; here we show that the complexity of decision variants of fixed-point problems, including Nash equilibria, are complete for this class, complementing work by Etessami and Yannakakis [13].
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