### Rigidity and polynomial invariants of convex polytopes

Maksym Fedorchuk and Igor Pak
Source: Duke Math. J. Volume 129, Number 2 (2005), 371-404.

#### Abstract

We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of the Robbins conjecture on the degree of generalized Heron polynomials.

First Page:
Primary Subjects: 52B10
Secondary Subjects: 51M20, 51M25, 52C25
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1127831442
Digital Object Identifier: doi:10.1215/S0012-7094-05-12926-X
Mathematical Reviews number (MathSciNet): MR2165546
Zentralblatt MATH identifier: 1081.52012

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