Abstract
We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast monomial transforms of triangular sets. Given a Gröbner basis, we prove that our algorithm requires only a polynomial number of arithmetic operations, and, for ideals in shape position, we show that its timings compare well against univariate factorization and backsubstitution. We conclude that the complexity of computing positive-dimensional tropical varieties via a traversal of the Gröbner complex is dominated by the complexity of the Gröbner walk.
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Acknowledgements
The authors would like to thank Avi Kulkarni (Dartmouth College) for his Magma script for solving polynomial equations over padic numbers, Marta Panizzut (TU Berlin) and Bernd Sturmfels (MPI MiS Leipzig + UC Berkeley) for the examples of tropical cubic surfaces with 27 distinct lines, as well as Andreas Steenpaß(TU Kaiserslautern) for technical support on the Singular library modular.lib (Steenpass 2019). The authors would also like to thank the anonymous referees for their comments and for pointing out a mistake in the previous version.
Paul Görlach and Yue Ren were partially supported by the Max Planck Institute for Mathematics in the Sciences. Paul Görlach acknowledges partial support by the DFG grant SE 1114/5-2, and Yue Ren is supported by the UKRI Future Leaders Fellowship (MR/S034463/1).
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Görlach , P., Ren, Y. & Zhang, L. Computing zero-dimensional tropical varieties via projections. comput. complex. 31, 5 (2022). https://doi.org/10.1007/s00037-022-00222-9
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DOI: https://doi.org/10.1007/s00037-022-00222-9