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On Read-k Projections of the Determinant

Authors Pavel Hrubeš , Pushkar S. Joglekar

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • determinant
  • permanent
  • projection of determinant
  • VNP completeness of permanent

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Abstract

We consider read-k determinantal representations of polynomials and prove some non-expressibility results. A square matrix M whose entries are variables or field elements will be called read-k, if every variable occurs at most k times in M. It will be called a determinantal representation of a polynomial f if f = det(M). We show that  
- the n × n permanent polynomial does not have a read-k determinantal representation for k ∈ o(√n/log n) (over a field of characteristic different from two).  We also obtain a quantitative strengthening of this result by giving a similar non-expressibility for k ∈ o(√n/log n) for an explicit n-variate multilinear polynomial (as opposed to the permanent which is n²-variate).

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Pavel Hrubeš and Pushkar S. Joglekar. On Read-k Projections of the Determinant. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 53:1-53:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.53

Author Details

Pavel Hrubeš
  • Institute of Mathematics of ASCR, Czech Republic
Pushkar S. Joglekar
  • Vishwakarma Institute of Technology, Pune, India

Funding

  • Hrubeš, Pavel: This work was supported by Czech Science Foundation GAČR grant 25-16311S.

References

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