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Mixed volume techniques for embeddings of Laman graphs

2009; Elsevier BV; Volume: 43; Issue: 2 Linguagem: Inglês

10.1016/j.comgeo.2009.04.004

1879-081X

Reinhard Steffens, Thorsten Theobald,

Advanced Numerical Analysis Techniques

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is to provide methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.