ER-completeness of isolated-point existence in semialgebraic sets

Ascertain whether deciding if a semialgebraic set contains an isolated point is ER-complete (as opposed to merely ER-hard).

Background

The isolated zero problem for algebraic sets is ER-complete, but for semialgebraic sets the corresponding problem (existence of an isolated point) has only ER-hardness known.

A completeness result would align the semialgebraic variant with its algebraic counterpart (ISO, H), strengthening the landscape of semialgebraic decision problems.

References

In and known to be -hard by B\"urgisser, CuckerCorollary 6.8, Corollary 9.4, but not known to be -complete.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Compendium — Problem 'Semialgebraic Set with Isolated Point (Open)'