K. Robin McLean

Groups of rational functions

In an article full of concrete examples, [1], Malcolm Perella asked what finite groups can be realised as groups of rational functions. When I first thought about this question, I imagined that most of the answers would be easy to locate in standard literature. Some of them are. But several have defied all my attempts to unearth them. I expect that they are hiding somewhere (as we said in our family when precious toys went astray), but they are certainly not in the books where I expected to find them! The present article attempts to answer Perella’s question by referring to appropriate sources where my search has succeeded, and supplying my own answers in other cases. Examples of the groups that arise are given in a series of exercises for readers.

The tiling conjecture for equiangular polygons

In [1], Derek Ball made three conjectures about equiangular polygons in which the length of each side is an integer. He called these integer equiangular polygons. The first two conjectures were proved in an earlier note, and the third is proved here. It can be stated as follows. The tiling conjecture : For each integer n ⩾ 3, there is a finite set T n of tiles such that every integer equiangular n -gon can be tiled by sufficiently many congruent copies of tiles in T n .

Cyclotomic and double angle polynomials

One admires and applauds the enterprise of anyone who uses Gauss’s 1801 Disquisitiones arithmeticae as the starting point for mathematical exploration. I enjoyed McKeon and Sherry’s description of their journey [1] and the challenge of their conjectures. They drew attention to a class of polynomials that satisfy what they called the double angle condition ((1) below). Unfortunately, their failure to work with an appropriate definition of cyclotomic polynomials seriously handicapped their computer-aided attempt to classify double angle polynomials. Once this is remedied, a pleasant classification emerges, at least for polynomials with rational coefficients, without recourse to a computer. The main aim of this article is to present this classification. A brief final section considers McKeon and Sherry’s conjectures about irreducible double angle polynomials.