RIGIDITY IN m - SPACE JACK

Matroid theory has many classical applications in combinatorics and geometry. Abstract rigidity matroids are generalizations of the infinitesi-mal rigidity matroids of frameworks in Euclidean space. In this paper we give a local characterization for abstract rigidity in any dimension. The conditions in this characterization are in many instances easier to verify than those in the definition of these matroids. A framework in m-space is a triple (V, E, p), where (V, E) is a graph and p is an embedding of V into real m-space. Let V = {1, 2,. .. , n}. Regarding p as a point in real mn-space, the distance constraints corresponding to E give a system of |E| quadratic equations in the coordinates of R mn. The solution set of these equations is an algebraic set A in mn-space called the configuration space of p. Clearly p ∈ A and we may describe a physical movement of the framework in space, that is, a movement of the vertices which preserves the lengths of the edges, by a path in A starting at p. A framework is rigid if the points on A in a neighborhood of p all correspond to a framework congruent to (V, E, p), i.e. the only motions which preserve all of the lengths of the edges are the direct isometries of R m. One approach to detecting rigidity is to replace the system of quadratic equations with their derivatives,