Sturm's 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. The only work in this connection that I managed to find is by Itenberg and Roy, who postulated a conjectural extension of Descartes' rule of signs, which however later was shown to be false.