I'm working on a problem in multiplicative ergodic theory, and Mahler measure has just made [*another*](http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf) appearance. I am looking for a uniform lower bound on Mahler measure over all polynomials of *fixed degree* with complex coefficients (not necessarily monic) where the largest coefficient is of absolute value 1. 


I found an ugly bound that is exponential in the degree, but am hoping for something in the literature or something a bit prettier. 



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> Can anyone give me a reference (or simple argument) for a lower bound for Mahler measure of degree $d$ polynomials (not necessarily monic) in a single variable (with complex coefficients) where the largest coefficient is of absolute value 1?
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Uniform Mahler Measure Lower Bound
I'm not sure I understand your question. $M(f)\ge1$ for all $f$, and if you take $f(x)=x^d+x^{d-1}+\cdots+1$, them $M(f)=1$, since $f$ is a product of cyclotomic polynomials. So you can't improve on the lower bound of $1$. If you want to exclude polynomials containing cyclotomic factors, then it's not clear that assuming the largest coefficient has absolute value $1$ will help you. For polynomials with integer coefficients, there is a bound of the form $M(f) \ge C (\log d)/(\log\log d)^{-3}$ due to Dobrowolski, but you probably already know that. OTOH, if you're really going to allow arbitrary complex coefficients, then I'm surprised you can get a bound even for non-cyclotomics.


ADDENDUM for non-monic polynomials


Let $|f|$ denote the magnitude of the largest coefficient of $f\in\mathbb C[x]$. There's a non-trivial upper bound for the Mahler measure 
$$ M(f) \le |f|\cdot (\deg f + 1)^{1/2}, $$
see for example *Fundamentals of Diophantine Geometry*, Serge Lang, Springer, Chapter 3, Theorem 2.8. (I realize you're asking for a lower bound.) The lower bound provided by the same theorem decays exponentially,
$$ M(f) \ge 2^{-\deg f}|f|. $$
It's not clear you can do too much better than that. For example,
$$f(x) = \frac{(x+1)^{2d}}{\binom{2d}{d}}$$ has largest coefficient $1$ and,
since $M((x+1)^d)=M(x+1)^d=1$, it has Mahler measure
$$ M(f) = \frac{1}{\binom{2d}{d}} \asymp \frac{\sqrt{d}}{4^d} \asymp \frac{\sqrt{\deg f}}{2^{\deg f}}.$$
In any case, you can't hope for better than an exponential lower bound.


