# Unambiguous Problems and Classes over Reals

Are there unambiguous analogues of $$NP_{R}$$ (using the BSS model, in all discussion)complete problems, and any results known about them? For instance, the canonical $$NP_{R}$$ complete problem $$4FEAS$$ ( ref. here but there could be a better reference ) could have an unambiguous version - $$U4FEAS$$ ( not allowing degenerate roots, for simplicity)in which there is only one zero or root for a system of degree 4 polynomial (for more about quantifiers over reals ). Of interest to study such unambiguous classes could be (loosely) Smales' 17th problem. How would the complexity differ (from say, $$NP_{R}$$, or higher in the $$PH_{R}$$ hierarchy) when we just have one root for a system of polynomials in the above context?

As the $$PH$$ can be defined over arbitrary structures - $$K$$, there is a $$PH_{K}$$, over that structure. We could have an Unambiguous Polynomial Hierarchy over the structure $$K$$- $$UPH_{K}$$- ( analogous Unambiguous Polynomial Hierarchy), are there any results of Unambiguous versions of classes over Reals?