Complexity, exactness, and rationality in polynomial optimization

We focus on rational solutions or nearly-feasible that serve as certificates of feasibility for polynomial optimization problems. show that, under some separability conditions, certain cubic polynomially constrained sets admit solutions. However, we in other cases it is NP Hard to detect if exist they any reasonable size. extend this idea various settings including near feasible, but super optimal and detecting rays which a function unbounded. Lastly, fixed dimension, the problem over set defined by inequalities providing simple certificate verify feasibility. conclude with several related examples irrationality encoding size issues QCQPs SOCPs.