Location of #R with respect to classical complexity classes

Characterize the counting class #R (the constant-free Boolean part of #P_R in the BSS model) with respect to traditional discrete complexity classes, including whether #R coincides with or separates from known counting classes and where it sits in the classical landscape.

Background

Bürgisser and Cucker introduced #R (denoted \BP(#P_\mathbb{R}0)) and proved completeness results for counting points in semialgebraic sets and computing Euler–Yao characteristics under various reductions. However, the exact placement of #R among classical discrete counting classes remains unknown.

Resolving this would refine our understanding of real-number counting complexity and its connections to discrete counting.

References

A counting version of was introduced by B\"urgisser and Cucker as $\BP(#P_{}0)$, which we could call $#$; they show that counting the number of points in a semialgebraic set is complete for this class, and computing the Euler-Yao characteristic of a semialgebraic set is complete for this class under Turing reductions. The paper leaves many interesting open questions, not least of it being the question of where $#$ is located with respect to traditional complexity classes.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Section 'What is next for ER?' — Counting Complexity