Undergraduate Research Project
Mahler Measure of Polynomials of Small Degree
Dan Madden 621-4665 Math 704 madden@math.arizona.edu
Bill McCallum 621-6697 Math 713 wmc@math.arizona.edu

Let f(x)=a_nxⁿ+a_n-1x^n-1+...+a₁x+a₀ be a polynomial with integer coeficients. The Mahler measure of such a polynomial can be defined in several equivalent ways, but perhaps the easiest is to consider its roots. Write

$$f(x)=a_n(x-\alpha_1)(x-\alpha_2).....(x-\alpha_n)$$

Some of these roots may be real numbers others come as complex conjugate pairs. To compute Mahler measure: we plot these roots on the complex plane: we eliminate all the roots on or in the unit circle; we multiply the absolute value of $a_n\,$together with the product of the absolute values of the remaining roots. In a formula, this is

$$M(f(x))=\vert\,a_n\vert\,\cdot\,\mbox{max\{1,}\vert\,\alpha_1... ...ert\mbox{\}}\cdot\mbox{max\{1,}\,\vert\,\alpha_n\vert\mbox{\}}$$

As we said, there are many equivalent, more natural, definitions of Mahler measure. Because of this there are many diverse applications for this measure. Nevertheless, Mahler measure is not very well understood.$\,\,$Here are a few quick results:

$1.\,M(f(x)\,g(x))=M(f(x))\,M(g(x))$

$2.\,M(\,f(-x))=M(f(x))$

$3.\,M(x^nf(\frac{1}{x}))=M(f(x))$

The principle mystery about Mahler measure begins with the fact that the polynomial

x¹⁰+x⁹-x⁷-x⁶-x⁵-x⁴-x³+x+1

has the smallest known Mahler measure that is not 1. Lehmer's conjecture states that this might actually be an example of the smallest possible Mahler measure bigger than 1. More precisely, Lehmer's conjecture is

$$\mbox{There is a number }C\mbox{ so that }M(f(x))<C\,\mbox{implies }M(f(x))\mbox{=1. }$$

There are some more interesting theorems about Mahler measure that might be valuable in proving the conjecture. For example

4. If M(f(x))=1, then $f(x)\,$is a factor of x^k-1 for some k.

5. If $f(x)\neq x\,$is irreducible and not a factor of x^k-1 for any k, then M(f(x))<1.4 implies that the coeficients of $f(x)\,$are symmetric, the same forward or backward.

There have been several numerical searches for polynomials with small Mahler measure, and for small degree the searches are exhaustive. Therefore the smallest Mahler for each degree up to 24 is known. The research project proposed here is to repeat this work, as far as possible, without resorting to a search through large numbers of polynomials. We propose starting with the smallest possible degrees, and providing theoretical proofs that the known lowest bounds are correct.