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<p>For every graph that is minimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. In this paper we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.</p>

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On Galois Groups of Type-1 Minimally Rigid Graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月11日

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  • Mehdi Makhul,
  • Josef Schicho,
  • Audie Warren

摘要

For every graph that is minimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. In this paper we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.