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Numerical Computation of Galois Groups

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Abstract

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

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Notes

  1. The defining polynomial of degree 12 was computed in [32, Ex. 6] and is available at the Web site https://sites.google.com/site/rootclassification/publications/DD.

  2. http://home.uchicago.edu/~joisro/quickLinks/NCGG/.

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Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jonathan D. Hauenstein

  2. Department of Statistics, University of Chicago, Chicago, IL, 60637, USA

    Jose Israel Rodriguez

  3. Department of Mathematics, Texas A & M University, College Station, TX, 77843, USA

    Frank Sottile

Authors
  1. Jonathan D. Hauenstein
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  2. Jose Israel Rodriguez
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  3. Frank Sottile
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Corresponding author

Correspondence to Jose Israel Rodriguez.

Additional information

Communicated by J E Cremona.

Research of Hauenstein supported in part by NSF Grant ACI-1460032, Sloan Research Fellowship BR2014-110 TR14, and Army Young Investigator Program (YIP) W911NF-15-1-0219.

Research of Rodriguez supported in part by NSF Grant DMS-1402545.

Research of Sottile supported in part by NSF Grant DMS-1501370.

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Hauenstein, J.D., Rodriguez, J.I. & Sottile, F. Numerical Computation of Galois Groups. Found Comput Math 18, 867–890 (2018). https://doi.org/10.1007/s10208-017-9356-x

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  • DOI: https://doi.org/10.1007/s10208-017-9356-x

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