Conjecture 4.1 — Lehmer's Problem

Definitions and mathematical statement

For a nonzero algebraic number $\alpha$ of degree $d$, let \[ f(X)=a_d\prod_{j=1}^d(X-\alpha_j)\in\mathbb Z[X] \] be its primitive irreducible polynomial with positive leading coefficient. The $\alpha_j$ are its complex conjugates and $\alpha_1=\alpha$ after a suitable ordering. Define the multiplicative Mahler measure and absolute logarithmic height by \[ M(\alpha)=a_d\prod_{j=1}^d\max\{1,|\alpha_j|\}, \qquad h(\alpha)=\frac{\log M(\alpha)}d. \] A root of unity is a complex number satisfying $\alpha^m=1$ for some integer $m\ge1$.

Does there exist an absolute constant $c>0$ such that $M(\alpha)\ge1+c$ for every nonzero algebraic number which is not a root of unity? The constant must be independent of its degree, conjugates, and defining polynomial.

Equivalently, does a constant $c'>0$ exist such that $h(\alpha)\ge c'/d$ for every such number of degree $d$? One may also state this for numbers of degree at most $d$, with the same uniform constant. The constants in these equivalent formulations need not be identical.

The leading coefficient is included so that the definition covers algebraic numbers which are not algebraic integers. Zero is excluded separately, and roots of unity must be excluded because their measure is one. The question asks for a uniform gap above one across unbounded degrees, not a separate bound for each fixed degree.

Short English statement

Is the Mahler measure of every nonzero algebraic number that is not a root of unity separated from one by an absolute positive gap?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700046 (AMR-086-0046), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 4.1. Original source indexed by AMR-086-0046. https://arxiv.org/abs/math/0312440.