On the Polya permanent problem over finite fields

2010/03/09 by Грегор Долинар, Alexander Guterman, Dolinar, Gregor +5 · 1 citation
Computer Science · Mathematics · #15A15 #15A33 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1003.1984

openalex publication_date 2010/03/09 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let \FF be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of \FF is sufficiently large. We also give an example of non-bijective map when \FF is arbitrary and an example of a bijective map when \FF is infinite which do transform permanent into determinant. The developed technique allows us to estimate the probability of the permanent and the determinant of matrices over finite fields to have a given value. Our results are also true over finite rings without zero divisors.

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• Permanental rank versus determinantal rank of random matrices over finite fields

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