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?
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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. 

