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Matrix Rigidity Depends on the Target Field

Authors László Babai , Bohdan Kivva

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Author Details

László Babai
  • University of Chicago, IL, USA
Bohdan Kivva
  • University of Chicago, IL, USA

Funding

The work of the authors has been partially supported by L. Babai’s NSF Grant CCF 1718902. All statements made are solely due to the authors and have not been evaluated or endorsed by the NSF.

Acknowledgements

The senior author thanks Zeev Dvir and Allen Liu for clarifications regarding their results. The authors thank an anomymous reviewer for pointing out the work of Samorodnitsky et al.

Cite AsGet BibTex

László Babai and Bohdan Kivva. Matrix Rigidity Depends on the Target Field. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 41:1-41:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.41

Abstract

The rigidity of a matrix A for target rank r is the minimum number of entries of A that need to be changed in order to obtain a matrix of rank at most r (Valiant, 1977). We study the dependence of rigidity on the target field. We consider especially two natural regimes: when one is allowed to make changes only from the field of definition of the matrix ("strict rigidity"), and when the changes are allowed to be in an arbitrary extension field ("absolute rigidity"). We demonstrate, apparently for the first time, a separation between these two concepts. We establish a gap of a factor of 3/2-o(1) between strict and absolute rigidities. The question seems especially timely because of recent results by Dvir and Liu (Theory of Computing, 2020) where important families of matrices, previously expected to be rigid, are shown not to be absolutely rigid, while their strict rigidity remains open. Our lower-bound method combines elementary arguments from algebraic geometry with "untouched minors" arguments. Finally, we point out that more families of long-time rigidity candidates fall as a consequence of the results of Dvir and Liu. These include the incidence matrices of projective planes over finite fields, proposed by Valiant as candidates for rigidity over 𝔽₂.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Complexity theory and logic
Keywords
  • Matrix rigidity
  • field extension

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References

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