Abstract.
In Valiant’s theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field \({\mathbb{F}}\) of characteristic ≠ 2, and an integer n, what is the minimum m such that the permanent of an n × n matrix X = (x ij ) can be expressed as a determinant of an m × m matrix, where the entries of the determinant matrix are affine linear functions of x ij ’s, and the equality is in \({\mathbb{F}}[{\bf X}]\). Mignon and Ressayre (2004) proved a quadratic lower bound \(m = \Omega(n^{2})\) for fields of characteristic 0. We extend the Mignon–Ressayre quadratic lower bound to all fields of characteristic ≠ 2.
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Manuscript received 24 December 2008
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Cai, JY., Chen, X. & Li, D. Quadratic Lower Bound for Permanent Vs. Determinant in any Characteristic. comput. complex. 19, 37–56 (2010). https://doi.org/10.1007/s00037-009-0284-2
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DOI: https://doi.org/10.1007/s00037-009-0284-2