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Constant-Depth Circuits for Polynomial GCD over Any Characteristic

Authors Somnath Bhattacharjee , Mrinal Kumar , Shanthanu S. Rai , Varun Ramanathan , Ramprasad Saptharishi , Shubhangi Saraf

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LIPIcs.CCC.2026.16.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • algebraic circuits
  • polynomial greatest common divisor
  • symmetric polynomials
  • finite fields
  • constant-depth circuits

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Abstract

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews & Wigderson who showed such an upper bound over fields of zero or large characteristic. 
Our proofs are based on a recent work of Bhattacharjee, Kumar, Rai, Ramanathan, Saptharishi & Saraf that shows closure of constant depth algebraic circuits under factorization. On our way to the proof, we show that any n-variate symmetric polynomial P that has a small constant depth algebraic circuit can be written as the composition of a small constant depth algebraic circuit with elementary symmetric polynomials. This statement is a constant depth version of a result of Bläser & Jindal, who showed this for algebraic circuits of unbounded depth. As an application of our techniques, we also strengthen the closure results for factors of constant-depth circuits in the work of Bhattacharjee et al. over fields for small characteristic.

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Somnath Bhattacharjee, Mrinal Kumar, Shanthanu S. Rai, Varun Ramanathan, Ramprasad Saptharishi, and Shubhangi Saraf. Constant-Depth Circuits for Polynomial GCD over Any Characteristic. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.16

Author Details

Somnath Bhattacharjee
  • University of Toronto, Canada
Mrinal Kumar
  • Tata Institute of Fundamental Research, Mumbai, India
Shanthanu S. Rai
  • Tata Institute of Fundamental Research, Mumbai, India
Varun Ramanathan
  • University of Copenhagen, Denmark
Ramprasad Saptharishi
  • Tata Institute of Fundamental Research, Mumbai, India
Shubhangi Saraf
  • University of Toronto, Canada

Funding

  • Bhattacharjee, Somnath: Research partially supported by an NSERC Discovery Grant.
  • Kumar, Mrinal: Research Supported by the Department of Atomic Energy, Government of India, under project number RTI4014, and in part by research grants from Google Research, Premji Invest and SERB.
  • Rai, Shanthanu S.: Research Supported by the Department of Atomic Energy, Government of India, under project number RTI4014, TCS Research Fellowship and in part by research grants from Google Research, Premji Invest and SERB.
  • Ramanathan, Varun: This work was conducted primarily while Varun Ramanathan was a doctoral student at TIFR Mumbai, and was supported by the Department of Atomic Energy, Government of India, under project number RTI4014, and in part by research grants from Google Research, Premji Invest and SERB. Final revisions were completed as a postdoctoral researcher at the University of Copenhagen, Denmark, supported by the European Research Council (ERC) under grant agreement no. 101125652 (ALBA).
  • Saptharishi, Ramprasad: Research Supported by the Department of Atomic Energy, Government of India, under project number RTI4014, and in part by research grants from Google Research, Premji Invest and SERB.
  • Saraf, Shubhangi: Research partially supported by the McLean Award and an NSERC Discovery Grant.

Acknowledgements

The discussions leading to this work started when a subset of the authors were at the workshop on Algebraic and Analytic Methods in Computational Complexity (Dagstuhl Seminar 24381) at Schloss Dagstuhl, and continued when they met again during the HDX & Codes workshop at ICTS-TIFR in Bengaluru. We are thankful to the organizers of these workshops and to the staff at these centers for the wonderful collaborative atmosphere that facilitated these discussions. Varun Ramanathan is grateful to Srikanth Srinivasan and Amik Raj Behera at the University of Copenhagen for helpful discussions on the complexity of symmetric polynomials.

References

  1. Robert Andrews and Avi Wigderson. Constant-depth arithmetic circuits for linear algebra problems. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 2367-2386, 2024. Full version at http://arxiv.org/abs/2404.10839. URL: https://doi.org/10.1109/FOCS61266.2024.00138.
  2. Somnath Bhattacharjee, Mrinal Kumar, Shanthanu Rai, Varun Ramanathan, Ramprasad Saptharishi, and Shubhangi Saraf. Closure under factorization from a result of furstenberg. Electronic Colloquium on Computational Complexity (ECCC), TR25-084, 2025. URL: https://eccc.weizmann.ac.il/report/2025/084.
  3. Markus Bläser and Gorav Jindal. On the Complexity of Symmetric Polynomials. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), volume 124 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1-47:14, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.47.
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