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

Abdel-Ghaffar, Khaled A. S.

Linear Algebra Appl. 315, No. 1-3, 139-144 (2000).

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 \(\mathbb F_q\).

Reviewer: A. A. Bogush (Minsk)

MSC:

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

Keywords:

random power series; matrices; determinants; finite fields; probability distribution

References:

[1]	Andrews, G.E., The theory of partitions, (1976), Addison-Wesley Reading, MA · Zbl 0371.10001
[2]	Brent, R.P.; McKay, B.D., Determinants and ranks of random matrices over \( Zm\), Discrete math., 66, 35-49, (1987) · Zbl 0628.15010
[3]	Knuth, D.E., Subspaces, subsets, and partitions, J. combin. theory ser. A., 10, 178-180, (1971) · Zbl 0221.05024
[4]	Landsberg, G., Über eine anzahlbestimmung und eine damit zusammenhängende reihe, J. reine angew. math., 111, 87-88, (1893) · JFM 25.0430.01