Rigid graph theory is an active area with many open problems, especially regarding embeddings in R d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C d and S d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S 2 and C 3. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

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Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R d or on a sphere and which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings.In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in R 3 . One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration.Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in R 2 and R 3 , while in the case of S 2 we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 [[1]].