[Sturm's theorem](http://en.wikipedia.org/wiki/Sturm%27s_theorem) gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P\_1,\dotsc,P\_n\in \mathbb{R}[X\_1,\dotsc,X\_n]$ be a collection of $n$ polynomials such that there are only finitely many roots of $P\_1=P\_2=\dotsb=P\_n=0$. I want to be able to compute the number of roots in $[a,b]^n$. I do not care if the roots are counted with or without multiplicity. 


I would also be interested in upper bounds on the number of roots that is similar to [Descartes' rule of signs](http://en.wikipedia.org/wiki/Descartes%27_rule_of_signs). The only work in this connection that I managed to find is by [Itenberg and Roy](http://www.ams.org/mathscinet-getitem?mr=1422527), who postulated a conjectural extension of Descartes' rule of signs, which however later was [shown to be false](http://www.ams.org/mathscinet-getitem?mr=1614424).



Counting roots: multidimensional Sturm's theorem
To expand on David's answer, the bound given by Khovanskii's theorem is of the form $2^{\binom{N}{2}} (n+1)^N$ per quadrant (more or less). Incremental improvements on this bounds have been obtained <http://arxiv.org/abs/1010.2962> being the latest, but nothing revolutionary and we're nowhere near realistic bounds.


As far as I know, there have been no new attempts on a multivariable Descartes (something that would take signs into consideration) since Itenberg and Roy's paper, and it remains a major open problem in the area.


**Added Later:** As far as an *algorithm* is concerned, I don't think you can count without solving, in which case the standard one to use would be the Cylindrical Algebraic Decomposition pioneered by Collins. It is more or less based on Sturm and induction, but it is a *lot* more involved than the 1-dimensional case. For reference, I recommend the book by Basu Pollack and Roy [Algorithms in Real Algebraic Geometry](http://Cylindrical%20Algebraic%20Decomposition%20pioneered%20by%20Collins/) (free download).


(More on counting without solving: Marie-Françoise Roy mentions the problem of determining if a semi-algebraic set is non-empty *without* producing one point per connected components as one of the major open problems in algorithmic real algebraic geometry).


