Graeme Taylor, Lehmer’s conjecture for matrices over the ring of integers of some imaginary quadratic fields, Journal of Number Theory, Volume 132, Issue 4, April 2012, Pages 590-607, ISSN 0022-314X, 10.1016/j.jnt.2011.09.006.
(http://www.sciencedirect.com/science/article/pii/S0022314X11002289)

I used this talk as an opportunity to present some results that were only at the conjectural stage last time I spoke on the topic. I have been working with Gary Greaves on Lehmer’s problem for matrices over the Gaussian and Eisenstein integers; we believe that we have proved the conjecture for those, and are slowly assembling a paper to that effect.

The paper covers the classification of the cyclotomic matrices/graphs for four of the six rings I considered in my thesis, but there have been some improvements to the methods. In particular, the proof that any maximal cyclotomic graph over those rings has all vertices of weighted degree four has been substantially streamlined; and there’s an explicit proof that any cyclotomic graph is contained in a maximal one. A follow-up paper proving Lehmer’s conjecture for polynomials arising from such graphs over the same rings is in preparation.

I try to avoid technical details of proofs in my talks, and to make the slides intelligible even if you weren’t there, so if you just want the motivation for, or results of, my PhD work then this is probably the best place to look. For all the proofs in tedious detail, there’s my thesis itself. I’ve since come up with a much more compact proof of the results in Chapter 5, which has lead to this draft paper classifying all cyclotomic L-graphs for d=-15,-11,-7 and -2: it’s far more compact than the corresponding sections of my thesis, but perhaps at the price of readability!

If you’d just like to know more about Mahler measure and Lehmer’s problem in general, then I’d recommend this survey by Chris Smyth. For various records related to small Mahler measure, see Mossinghof’s tables.

I shall soon be delivering the final copies of my thesis to the University of Edinburgh, who will in time make it available in both hard copy and electronically. But for convenience, you can already obtain the pdf version here.

We generalise the study of cyclotomic matrices – those with all eigenvalues in the interval [-2,2] – from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined.

We introduce the notion of `4-cyclotomic’ matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. The six rings of integers for Q(√(d)) for d=-1, -2, -3, -7, -11, -15 give rise to examples not found in the rational-integer case; in four (d=-1, -2, -3, -7) we recover infinite families as well as sporadic cases.

For d=-15, -11, -7, -2, we demonstrate that a maximal cyclotomic graph is necessarily 4-cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.

You can find my slides here, and an audio recording may become available in the future- the conference was held in Toronto and Vancouver via videoconference (which worked well) so hopefully all the talks will be archived online.

Update 24/x/09: Audio of all talks at Fields (hence, including mine) can now be found on their website.

Having hit a bit of a wall trying to prove that a maximal cyclotomic matrix necessarily squares to 4I, I’ve been exploring related questions instead. For instance, it only took a couple of tweaks to my code to search for matrices that square to 3I instead of 4I; there turn out to be only finitely many, which isn’t particularly interesting. However, one of them is an 8 vertex cube which I recognised as ‘half’ the maximal cyclotomic graph S₁₆.

This got me thinking about the properties of graphs obtained by stitching together two graphs, and lead to an interesting construction. If M squares to nI, then by taking a second copy of its graph, negating and joining each vertex in the original to the corresponding one in the copy, you get a new matrix which squares to (n+1)I.

By iterating this process, many of the maximal cyclotomic graphs can be recovered; and since there are infinite families of maximal cyclotomic graphs, I can demonstrate infinitely many non-trivial integer matrices with all eigenvales of absolute value sqrt(n) for any integer n≥5 too.

A particularly nice example is the family of signed n-hypercubes generated by running this procedure on the ’1-cyclotomic’ graph consisting of just a line. The picture at the top of this post is of the 5th step, and I’ve put together an animation illustrating its construction:

No mathematical reason to stop at 5, it just gets harder and harder to draw these things!

For cyclotomic matrices over the rational integers, it turns out that all maximal examples satisfy the equation M²=4I. However, this is a side effect of the classification, rather than an independent result which can be used to arrive at that classification. Which is a problem, as (despite attempts to guard against it) my approach implicity assumes this condition…

Thus what I currently have is a classification not necessarily of all the maximal cyclotomic graphs, but of all the graphs whose matrix representation squares to 4I; no less substantial a piece of work, but a less significant result.

It’s still my belief that this is the full classification, but to prove that, I’ll need to demonstrate the equivalence of these two conditions (the reverse direction I already have, but that’s no help); or at least the weaker result that any cyclotomic graph with a vertex of weighted degree less than four admits a cyclotomic supermatrix, which is the hidden assumption in my existing algorithm. Suggestions welcome!