Rigidity and polynomial invariants of convex polytopes. (English) Zbl 1081.52012

Fedorchuk, Maksym; Pak, Igor

Duke Math. J. 129, No. 2, 371-404 (2005).

Using ideas from algebraic geometry and delicate calculations, the authors introduce an algebraic approach to problems of rigidity of convex polytopes in great generality by establishing polynomial invariants that satisfy polynomial relations in terms of squared edge-lengths of these polytopes. On this way constructions of simplicial convex polytopes from a given triangulation and given edge-lengths are described. Sharp upper and lower bounds on the degree of these polynomial relations are obtained, and a number of examples and special cases are studied (for regular bipyramids even explicit formulae for some of these relations are derived). Furthermore, the authors conclude with a proof of the Robbins conjecture on the degree of generalized Héron polynomials.

Reviewer: Horst Martini (Chemnitz)

MSC:

52B10	Three-dimensional polytopes
51M20	Polyhedra and polytopes; regular figures, division of spaces
51M25	Length, area and volume in real or complex geometry
52C25	Rigidity and flexibility of structures (aspects of discrete geometry)

Keywords:

polynomial invariants; rigidity; flexible polyhedron; Héron polynomial; polytope graph

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