Duke Mathematical Journal

Rigidity and polynomial invariants of convex polytopes

Maksym Fedorchuk and Igor Pak

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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.

Article information

Duke Math. J., Volume 129, Number 2 (2005), 371-404.

First available in Project Euclid: 27 September 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B10: Three-dimensional polytopes
Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 51M25: Length, area and volume [See also 26B15] 52C25: Rigidity and flexibility of structures [See also 70B15]


Fedorchuk, Maksym; Pak, Igor. Rigidity and polynomial invariants of convex polytopes. Duke Math. J. 129 (2005), no. 2, 371--404. doi:10.1215/S0012-7094-05-12926-X. https://projecteuclid.org/euclid.dmj/1127831442

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