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Solving decomposable sparse systems

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Abstract

Améndola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems admits such a decomposition if and only if the corresponding Galois group is imprimitive. When the Galois group is imprimitive, we consider the problem of computing an explicit decomposition. A consequence of Esterov’s classification of sparse polynomial systems with imprimitive Galois groups is that this decomposition is obtained by inspection. This leads to a recursive algorithm to compute complex isolated solutions to decomposable sparse systems, which we present and give evidence for its efficiency.

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Funding

Research of Sottile supported by grant 636314 from the Simons Foundation.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Taylor Brysiewicz

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Jose Israel Rodriguez

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

    Frank Sottile & Thomas Yahl

Authors
  1. Taylor Brysiewicz
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  2. Jose Israel Rodriguez
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  3. Frank Sottile
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  4. Thomas Yahl
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Correspondence to Taylor Brysiewicz.

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Brysiewicz, T., Rodriguez, J.I., Sottile, F. et al. Solving decomposable sparse systems. Numer Algor 88, 453–474 (2021). https://doi.org/10.1007/s11075-020-01045-x

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