I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try...
Consider a multi-variate polynomial $P(x)$, is it possible to find explicit parametrizations of the regions where $P(x)>0$? By explicit I mean explicit enough so that one could compute (even just numerically) an integral of some function over some of the regions where $P(x)<0$.
I am especially interested in understanding whether the knowledge of the Newton polytope of $P(x)$ helps in any way.
