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Galois groups of Schubert problems via homotopy computation


Authors: Anton Leykin and Frank Sottile
Journal: Math. Comp. 78 (2009), 1749-1765
MSC (2000): Primary 14N15, 65H20
DOI: https://doi.org/10.1090/S0025-5718-09-02239-X
Published electronically: February 25, 2009
MathSciNet review: 2501073
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Abstract: Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in $ \mathbb{C}^8$ meeting 15 fixed 5-planes non-trivially is the full symmetric group $ S_{6006}$.


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Additional Information

Anton Leykin
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street (M/C 249), Chicago, Illinois 60607-7045
Email: leykin@math.uic.edu

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: sottile@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02239-X
Keywords: Polynomial homotopy continuation, Schubert problem, Galois group
Received by editor(s): February 22, 2008
Received by editor(s) in revised form: June 14, 2008
Published electronically: February 25, 2009
Additional Notes: The authors were supported by the Institute for Mathematics and its Applications and Sottile by NSF grants CAREER DMS-0538734 and DMS-0701050
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.