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The determinant of random power series matrices over finite fields. (English) Zbl 0964.15016
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 \(\mathbb F_q\).

15A15 Determinants, permanents, traces, other special matrix functions
60C05 Combinatorial probability
15B33 Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
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