The determinant of random power series matrices over finite fields (Q1583721)

The set \(M_{n\times n}(F_q[[x]])\) of \(n\times n\)-matrices \(A\) with elements \(a_{ij}= \sum a_{ijt} x^t\in\mathbb F_q[[x]]\) \((t\geq 0)\) is considered where \(\mathbb F_q[[x]]\) denotes the ring of formal power series over the finite field \(\mathbb F_q\) of \(q\) elements. The probability distribution is studied of the coefficient \(d_t\) of the determinant \(\det(A)= \sum d_tx^t\) \((t\geq 0)\) if \(A\) is random, i.e. if the coefficients of the entries of \(A\) are statistically independent and uniform over the elements of the finite field \(\mathbb F_q\).