  The study of (minimally) rigid graphs is motivated by numerous applications,
mostly in robotics and bioinformatics. A major open problem concerns the number
of embeddings of such graphs, up to rigid motions, in Euclidean space. We
capture embeddability by polynomial systems with suitable structure, so that
their mixed volume, which bounds the number of common roots, to yield
interesting upper bounds on the number of embeddings. We focus on $\RR^2$ and
$\RR^3$, where Laman graphs and 1-skeleta of convex simplicial polyhedra,
respectively, admit inductive Henneberg constructions. We establish the first
lower bound in $\RR^3$ of about $2.52^n$, where $n$ denotes the number of
vertices. Moreover, our implementation yields upper bounds for $n \le 10$ in
$\RR^2$ and $\RR^3$, which reduce the existing gaps, and tight bounds up to
$n=7$ in $\RR^3$.
