is the question whether a polynomial is nonnegative on some semialgebraic set (equivalently, is it in the cone denerated by some polynomials in the field of rational functions) known to be decidable?
Assuming you're working over the real numbers (or at least over a realclosed field), the answer is yes. The question lies within the firstorder theory of realclosed fields, and that theory is decidable by an old theorem of Tarski. 

