Files
Abstract
Given any commutative ring R, a commutator of two n×n matrices over R has trace 0. In the first part of the dissertation, we study the converse: whether every n×n trace 0 matrix is a commutator. We show that over a Bézout domain with an algebraically closed quotient field, every trace 0 matrix is a commutator. We then show that if R is a regular ring with a large enough dimension, then there exist an n×n trace 0 matrix that is not a commutator. This improves on a result ofLissner by increasing the size of the matrix n allowed for a fixed R.
In the second part, we study cycles associated with the b-ary expansion of positive integers for some fixed b≥2. More specifically, if ϕ(x) is an integer polynomial such that ϕ(n)>0 for all n>0, then consider the map S:ℤ→ℤ, with S(n):=ϕ(x0)+… +ϕ(xd) where n=x0+x1 b+ … +xd b^d is the b-ary expansion of n. It is known that the orbit set {n, S(n), S(S(n)),…} reaches a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Fix now an integer ℓ ≥ 1 and let ϕ(x) = x². We show that the set of bases b≥2 which have at least one cycle of length ℓ always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that there may be infinitely many bases with exactly two cycles.