If VNP Is Hard, Then so Are Equations for It

Authors Mrinal Kumar, C. Ramya , Ramprasad Saptharishi , Anamay Tengse

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Author Details

Mrinal Kumar
  • Indian Institute of Technology Bombay, India
C. Ramya
  • Chennai Mathematical Institute, India
Ramprasad Saptharishi
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Anamay Tengse
  • Department of Computer Science, University of Haifa, Israel


We thank anonymous reviewers of an earlier version of this paper and Joshua Grochow, whose questions pointed us in the direction of this result. We also thank the reviewers of STACS 2022 for their helpful comments and suggestions. We also thank Prerona Chatterjee and Ben Lee Volk for helpful discussions at various stages of this work.

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Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse. If VNP Is Hard, Then so Are Equations for It. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee, Kumar, Ramya, Saptharishi and Tengse (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Computational Complexity
  • Algebraic Circuits
  • Algebraic Natural Proofs


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