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.