is the question whether a polynomial is non-negative on some semi-algebraic set (equivalently, is it in the cone denerated by some polynomials in the field of rational functions) known to be decidable?



Decidability of the generated order
Assuming you're working over the real numbers (or at least over a real-closed field), the answer is yes. The question lies within the first-order theory of real-closed fields, and that theory is decidable by an old theorem of Tarski. 


