Beyond the Existential Theory of the Reals

TL;DR

This work develops a robust, finite-level real hierarchy based on the theory of the reals, showing that key complexity levels are invariant under signature changes and domain bounds. It establishes foundational equalities such as $\Sigma_k^{\mathbb{R}} = \Sigma_k^{<\mathbb{R}}$ and $\Pi_k^{\mathbb{R}} = \Pi_k^{<\mathbb{R}}$ for all $k\ge1$, and extends these robustness results to bounded-open and, partially, bounded-closed universes. By introducing restricted complete problems (e.g., $\Sigma_k^{\#}$-POLY, $\Pi_k^{\#}$-POLY) and leveraging Solernó’s Łojasiewicz inequality along with Tseitin reductions, the paper sharpens previous results on semialgebraic-set properties and the Hausdorff-distance problem within the real hierarchy. The findings provide a unified framework for higher-level real-algebraic complexity, enabling more precise hardness reductions and guiding future work on exotic quantifiers and bounded-domain variants. The work has potential impact on computational real algebraic geometry, optimization, and related areas by clarifying the landscape of higher-level decision problems over the reals.

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 Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

Theorems & Definitions (59)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Conjecture 1.4
• Lemma 1.5
• Proposition 2.1
• proof
• Theorem 2.2: Solernó
• Theorem 2.3
• proof