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.
"Rigidity and polynomial invariants of convex polytopes." Duke Math. J. 129 (2) 371 - 404, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12926-X