TY - JOUR
T1 - Permanent Versus Determinant over a Finite Field
AU - Dolinar, G.
AU - Guterman, A.
AU - Kuzma, B.
AU - Orel, M.
N1 - Funding Information: Acknowledgements. This research was partially supported by a joint Slovene–Russian grant BI-RU/08-09-009. The research of the second author is also supported by the Russian Foundation for Basic Research (project No. 09-01-00303a).
PY - 2013/9
Y1 - 2013/9
N2 - Let F be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices (Formula Presented) and for the whole matrix space M n(F). It is known that for n = 2, there are bijective linear maps Φ on (Formula Presented) and M n(F) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, φ{symbol}), where Φ is an arbitrary bijective map on matrices and (Formula Presented) is an arbitrary map such that per A = φ{symbol}(det Φ(A)) for all matrices A from the spaces(Formula Presented) and M n(F), respectively. Moreover, for the space M n(F), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field F contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.
AB - Let F be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices (Formula Presented) and for the whole matrix space M n(F). It is known that for n = 2, there are bijective linear maps Φ on (Formula Presented) and M n(F) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, φ{symbol}), where Φ is an arbitrary bijective map on matrices and (Formula Presented) is an arbitrary map such that per A = φ{symbol}(det Φ(A)) for all matrices A from the spaces(Formula Presented) and M n(F), respectively. Moreover, for the space M n(F), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field F contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.
UR - http://www.scopus.com/inward/record.url?scp=84899425746&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s10958-013-1469-4
DO - https://doi.org/10.1007/s10958-013-1469-4
M3 - مقالة
SN - 1072-3374
VL - 193
SP - 404
EP - 413
JO - Journal of Mathematical Sciences
JF - Journal of Mathematical Sciences
IS - 3
ER -