Is there a way to determine exactly (without the use of approximation methods) whether $p\in \mathbb{R}[x_1,\dots,x_n]$ has realvalued solutions.
Algorithms based on Sturm's theorem seem to be applicable to univariate polynomials only.
Is there a way to determine exactly (without the use of approximation methods) whether $p\in \mathbb{R}[x_1,\dots,x_n]$ has realvalued solutions. Algorithms based on Sturm's theorem seem to be applicable to univariate polynomials only. 


Tarski's theorem on the decidability of the theory of realclosed fields provides a general algorithm that decides any question expressible in the first order language of realclosed fields. His algorithm can therefore determine, for any statement, whether it is true in the structure $\langle\mathbb{R},+,\cdot,0,1,\lt\rangle$. Thus, not only are the purely existential assertions (solvability of systems of equations) decidable in this context, but also more complex assertions involving iterated quantifiers, which would not seem without this result to be decidable even by approximation. The way Tarski's argument proceeds is by elimination of quantifiers: every assertion in this language is equivalent to a quantifierfree assertion. In particular, the existence of a solution to $p(\vec x)=0$ is equivalent by Tarski's reduction to a quantifierfree assertion about the coefficients of the polynomial. That is, the algorithm reduces the question to a mere calculation involving the coefficients. But if you are interested in actually using the algorithm in specific instances, rather than the theoretical question about whether in principal there is such an algorithm, then Tarski's algorithm may not actually be helpful. Although it has been implemented on computers, the algorithm takes something like a tower of exponential time in the size of the input, and evidently it has been proved that every quantifierelimination algorithm must be at least doubleexponential. 


This problem is solved in socalled Semialgebraic Geometry. Here are some books: Basu S. Algorithms in Semialgebraic Geometry Basu S., Pollack R., Roy M.F. Algorithms in Real Algebraic Geometry Bochnak J., Coste M., Roy MF. Real algebraic geometry Coste M. An introduction to semialgebraic geometry 

