Permanent Versus Determinant over a Finite Field

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.