On the Polya permanent problem over finite fields

Authors:
Gregor Dolinar;Alexander E. Guterman;Bojan Kuzma;Marko Orel
Affiliations:
Faculty of Electrical Engineering, University of Ljubljana, Traška 25, SI-1000 Ljubljana, Slovenia;Faculty of Algebra, Department of Mathematics and Mechanics, Moscow State University, GSP-1, 119991 Moscow, Russia;University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia and IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia;IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia
Venue:
European Journal of Combinatorics
Year:
2011

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Abstract

Let F be a finite field of characteristic different from 2. We show that no bijective map transforms the permanent into the determinant when the cardinality of F is sufficiently large. We also give an example of a non-bijective map when F is arbitrary and an example of a bijective map when F is infinite which do transform the permanent into the determinant. The technique developed allows us to estimate the probability of the permanent and the determinant of matrices over finite fields having a given value. Our results are also true over finite rings without zero divisors.