On the distribution of permanents of matrices over finite fields

For a prime power q, we study the distribution of permanents of n × n matrices over a finite field F q of q elements. We show that if A is a sufficient large subset of F q then the set of permanents of n × n matrices with entries in A covers all (or almost) F q ∗ . When q = p is a prime, and A is a subinterval of [ 0 , p − 1 ] of cardinality | A | ≫ p 1 / 2 log p , we show that the number of matrices with entries in A having permanent t is asymptotically close to the expected value. We also study this problem in more general settings.