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Algebraic detection of $T_k$-configurations
Carl Friedrich Kreiner, Johannes Zimmer and Isaac Chenchiah
The problem of detecting so-called $T_k$-configurations is addressed here. These configurations are the most prominent examples of sets with nontrivial rank-one convex hulls. Rank-one convex hulls play an important rôle in the calculus of variations and the modelling of effective properties of materials.
An efficient algorithm, based on algebraic methods, is presented. Unlike previous work on the computation of rank-one convex hulls, it is not based on discretization and gives exact results. This algorithm enables, for the first time, large numbers of tests for these configurations. Stochastic experiments in several space dimensions are reported here.