I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials.
I thought I would project the solutions onto $x$ and $y$ axis by means of resultant computations. Then I would isolate the roots of two 9th degree univariate polynomials which would give me at most 9 $\times$ 9 candidate regions.
But then I got stuck: how do I know for sure which regions do contain the roots, and which do not? Is it sufficient (i.e. is there such a test) to exclude all the regions which do not contain any roots or do I also need some kind of "inclusion predicate" to be really sure I found the right regions?
To put it differently: how do I "match" the isolating intervals of one univariate polynomial ($x$) with the intervals of the other univariate polynomial ($y$) so that the pair demarcates a region having a solution of the original system?