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We use Bernstein's Theorem [1] to obtain combinatorial bounds for the number of embeddings of Laman graph frameworks modulo rigid motions. For this, we study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. The bounds can easily be computed and for some classes of graphs, the bounds are tight.

Let g 1 ,. .. , g k be tropical polynomials in n variables with Newton polytopes P 1 ,. .. , P k. We study combinatorial questions on the intersection of the tropical hyper-surfaces defined by g 1 ,. .. , g k , such as the f-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work [33] who… (More)

- Reinhard Johannes Steffens, Vom Fachbereich, Johann Wolfgang
- 2009

The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer… (More)

We give some new technical tools which simplify mixed volume computation for larger polynomial systems and allow the computation of mixed volume bounds for polynomial systems of arbitrary dimension arising in various applications as seen in [14].

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