Beyond the Existential Theory of the Reals

TL;DR

This paper explores the robustness of higher-level completeness in the theory of the reals, strengthening the hierarchy and identifying new complete problems with implications for computational geometry and complexity.

Contribution

It demonstrates that higher-level completeness is stable under various conditions and introduces new complete problems for the real hierarchy.

Findings

Established robustness of higher-level completeness in the theory of the reals.

Identified several new families of complete problems.

Sharpened complexity results for properties of semialgebraic sets.

Abstract

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 B\"{u}rgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.