We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory, we give a proof of a result of Lieb, Jordan and Helmke on the number of linear unimodular matrix polynomials over a finite field. As an application of our results, we give a new proof of a theorem of Chen and Tseng which answers a question of Niederreiter on splitting subspaces. We use our results to affirmatively resolve a conjecture on the probability that a matrix polynomial is unimodular.

中文翻译：

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有限域上的单模多项式矩阵

我们考虑有限域上矩阵多项式的一些组合问题。使用控制理论的结果，我们证明了Lieb，Jordan和Helmke关于有限域上线性单模矩阵多项式的数目的结果。作为我们结果的应用，我们给出了Chen和Tseng定理的新证明，该定理回答了Niederreiter关于分裂子空间的问题。我们使用我们的结果来肯定地解决矩阵多项式为单模概率的猜想。