The existential theory of the reals (ETR) consists of exis-tentially quantified boolean formulas over equalities and inequalities of real-valued polynomials. We propose the approximate existential theory of the reals (Є-ETR), in which the constraints only need to be satisfied approximately. We first show that unconstrained Є-ETR = ETR, and then study the Є-ETR problem when the solution is constrained to lie in a given convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula. A consequence of our theorem is that we obtain a quasi-polynomial time approximation scheme (QPTAS) for a fragment of constrained Є-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.
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