极小刚性图嵌入数的多重齐次Bézout界
On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs
摘要：
刚性图论是一个活跃的领域，有许多开放的问题，特别是关于嵌入在$\mathbb{R}^d$或其他流形，以及严格的上限，他们的数量为一个给定的顶点数。我们的前提是将嵌入的数量与一个约束良好的代数系统的解的数量联系起来，并利用后一领域的进展。特别是，该系统的复杂的解决方案，自然地扩展了实嵌入的概念，从而使我们能够采用复根的界限。我们专注于多齐次B{é}zout（m-B{é}zout）界的代数系统，因为他们是快速计算和相当紧的系统表现出的结构，在我们的情况下。我们介绍了两种方法，这样的界限有关的组合性质的极小刚性图在$\mathbb{C}^d$和$S^d$。第一个关系到的m-Bézout界的图形方向的数量，而第二个利用矩阵永久制定。使用这些方法，我们提高了最好的已知的渐近上界的可平面图在3维，和所有极小刚性图在维度$d\geq 5$，无论是在欧几里德和球形的情况下。我们的计算表明，可平面图在$S^2$和$\mathbb{C}^3$中的嵌入的m-Bézout界是紧的.利用Bernstein关于混合体积精确性的第二定理，通过分析相应的Newton多面体，将其与m-B{é}zout界联系起来.我们通过指数因子减少了验证准确性所需的检查次数，并进一步推测，检查线性而不是指数数量的情况就足够了。
英文摘要：
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{é}zout (m-B{é}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-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 $d\geq 5$, 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 $\mathbb{C}^3$. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-B{é}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.
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