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Complexity of Linear Circuits and Geometry

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Abstract

We use algebraic geometry to study matrix rigidity and, more generally, the complexity of computing a matrix–vector product, continuing a study initiated in Kumar et al. (2009), Landsberg et al. (preprint). In particular, we (1) exhibit many non-obvious equations testing for (border) rigidity, (2) compute degrees of varieties associated with rigidity, (3) describe algebraic varieties associated with families of matrices that are expected to have super-linear rigidity, and (4) prove results about the ideals and degrees of cones that are of interest in their own right.

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Acknowledgments

We thank the anonymous referees for very careful reading and numerous useful suggestions.

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Authors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Fulvio Gesmundo, Christian Ikenmeyer & J. M. Landsberg

  2. University of Notre Dame, Notre Dame, IN, USA

    Jonathan D. Hauenstein

Authors
  1. Fulvio Gesmundo
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  2. Jonathan D. Hauenstein
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  3. Christian Ikenmeyer
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  4. J. M. Landsberg
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Corresponding author

Correspondence to J. M. Landsberg.

Additional information

Communicated by Stephen Cook.

Hauenstein was supported by NSF Grant DMS-1262428 and DARPA Young Faculty Award (YFA). Landsberg supported by NSF Grant DMS-1006353.

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Gesmundo, F., Hauenstein, J.D., Ikenmeyer, C. et al. Complexity of Linear Circuits and Geometry. Found Comput Math 16, 599–635 (2016). https://doi.org/10.1007/s10208-015-9258-8

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  • DOI: https://doi.org/10.1007/s10208-015-9258-8

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