On the multihomogeneous B\'ezout bound on the number of embeddings of minimally rigid graphs

TL;DR

This paper introduces new methods using multihomogeneous Bézout bounds to estimate the number of embeddings of minimally rigid graphs, improving bounds in certain dimensions and relating algebraic solutions to graph properties.

Contribution

It develops two novel approaches connecting algebraic bounds to graph orientations and matrix permanents, leading to tighter upper bounds for graph embeddings in various dimensions.

Findings

Improved asymptotic upper bounds for planar graphs in dimension 3.

Tightness of m-Bézout bounds for embeddings of planar graphs in S^2 and C^3.

Reduced computational checks for solution exactness using Bernstein's theorem.

Abstract

Rigid graph theory is an active area with many open problems, especially regarding embeddings in $\mathbb{R}^d$ 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 B{\'e}zout (m-B{\'e}zout) 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 $\mathbb{C}^d$ and $S^d$. The first relates the number of graph orientations to the m-B\'ezout 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 $d\geq 5$, both in the Euclidean and spherical case. Our computations indicate that m-B\'ezout bounds are tight for embeddings of planar graphs in $S^2$ and $\mathbb{C}^3$. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-B{\'e}zout 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.