Lehmer’s Mahler measure problem

A polynomial’s roots are numbers that make it zero. For $x^2-3x+1$, they are about 2.618 and 0.382. Replace each root’s size by 1 if it is smaller than 1, then multiply: here the result is 2.618. The leading coefficient is 1, so this product is its Mahler measure. Lehmer asks whether every nonzero polynomial with whole-number coefficients has measure either exactly 1 or at least a fixed amount above 1.[1]

Compute Mahler measure by multiplying the leading-coefficient size and the sizes of all roots outside the unit circle; other roots contribute 1. For integer coefficients, every factor in $M(P)$ is at least 1. If the leading coefficient has size at least 2, then $M(P)\ge2$, so measures just above 1 can occur only with leading coefficient $\pm1$. Changing a polynomial’s sign leaves its measure unchanged. Thus the gap for all nonzero integer polynomials is equivalent to the gap for monic polynomials, whose leading coefficient is 1.[1]

The same gap must work for polynomials with any number of roots. Checking quadratics or even a vast computer list cannot give a bound for them all. A bound that weakens as the number of roots grows also leaves the question open.[1]

The target is a single $\lambda>1$ with $M(P)=1$ or $M(P)\ge\lambda$, regardless of degree. A lower bound that approaches 1 as degree increases cannot supply this constant. The March 2026 bivariate-family paper discusses the continuing Lehmer question; its multivariable results do not prove this universal one-variable statement.[1] [2]

Roots also describe stretching in certain reversible motions defined by whole-number coefficients on a space with opposite edges identified. Mahler measure determines the growth of distinguishable motion histories. A uniform gap would put a minimum on positive complexity growth for this particular family, across every dimension.[1] [3]

Identify coordinate vectors that differ by integers to form a torus, $\mathbb{R}^d/\mathbb{Z}^d$. An integer matrix $A$ with determinant $\pm1$ defines a reversible map $T_A$ on it. For its characteristic polynomial $P(x)=\det(xI-A)$, the exact relation is $h_{\mathrm{top}}(T_A)=\log M(P)$. Lehmer’s gap would therefore bound every positive entropy in this family below by $\log\lambda$, independently of dimension. The arithmetic restriction on coefficients would impose a quantitative restriction on dynamical complexity.[1] [3]