Foundations of Computational Mathematics

, Volume 16, Issue 3, pp 599–635 | Cite as

Complexity of Linear Circuits and Geometry

  • Fulvio Gesmundo
  • Jonathan D. Hauenstein
  • Christian Ikenmeyer
  • J. M. LandsbergEmail author


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.


Matrix rigidity Discrete Fourier transform Vandermonde matrix Cauchy matrix 

Mathematics Subject Classification

68Q17 15B05 65T50 



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


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© SFoCM 2015

Authors and Affiliations

  • Fulvio Gesmundo
    • 1
  • Jonathan D. Hauenstein
    • 2
  • Christian Ikenmeyer
    • 1
  • J. M. Landsberg
    • 1
    Email author
  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.University of Notre DameNotre DameUSA

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