Unimodular polynomial matrices and partial linear transformations over finite fields

Abstract

This thesis consists of some interesting combinatorial problems on matrix poly- nomials over finite fields. Using results from control theory, we give a proof of a result of Lieb, Jordan and Helmke (2016) which solves the problem of counting the number of linear unimodular polynomial matrices over a finite field. This problem was essentially considered by Kocięcki and Przyłuski in an attempt to estimate the proportion of reachable linear systems over a finite field. As an appli- cation 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. We consider another enumerative problem on the similarity class of an arbitrary linear map defined on a subspace of a vector space over a finite field. Let V be a finite-dimensional vector space over the finite field Fq with q elements where q is a prime power and suppose W and Wf are subspaces of V . Two linear transformations T : W → V and Te : Wf → V are said to be similar if there exists a linear isomorphism S : V → V such that the following diagram commutes: W V Wf V T SW ≃ S Te . In other words, we must have S ◦T = Te◦SW where SW denotes the restriction of S to W. Given a linear map T defined on a subspace W of V , sometimes referred to as a partial linear map, we discuss the similarity invariants for T. We then give an explicit formula for the number of linear maps that are similar to T. The case where T is a linear operator on V (the case W = V ) is well-studied and we extend the result where W is an arbitrary subspace of V . In fact, the problem of counting the similarity class size of a linear operator T is equivalent to counting the number of square matrices over the finite field Fq in a conjugacy class. This problem has been studied by Kung (1981) and Stong (1988) among others, and an explicit formula due to Philip Hall is known. Our results extend the explicit formula of Philip Hall on matrix conjugacy class size. As a consequence of the results, we provide another proof of a theorem of Lieb, Jordan and Helmke on the number of linear unimodular matrix polynomials.