We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

中文翻译：

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超越实数的存在理论

我们证明，实数理论的更高层次上的完整性是一个稳健的概念（在改变签名和限制量词的域的情况下）。这弥补了层次结构中公认的差距，并为各种计算问题带来了更强的完整性结果。我们展示了几个完整问题系列，可用于真实层次结构中未来的完整性结果。作为一个应用，我们强化了 Bürgisser 和 Cucker 关于半代数集性质复杂性的一些结果，包括 Jungeblut、Kleist 和 Miltzow 也研究过的 Hausdorff 距离问题。