LIPIcs.STACS.2025.53.pdf
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On Read-k Projections of the Determinant
Authors
Pavel Hrubeš 
,
Pushkar S. Joglekar
-
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42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Theoretical Aspects of Computer Science (STACS) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2025-02-24
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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
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).
Subject Classification
ACM Subject Classification
- Theory of computation → Algebraic complexity theory
Keywords
- determinant
- permanent
- projection of determinant
- VNP completeness of permanent
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References
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