Entropy values and Lehmer's problem

Definitions and mathematical statement

For a monic polynomial $f\in\mathbb Z[x]$ of positive degree $d$, write $f(x)=\prod_{j=1}^d(x-\alpha_j)$, listing complex roots with multiplicity. Its logarithmic Mahler measure is \[ m(f)=\sum_{j=1}^d\log\max\{1,|\alpha_j|\}, \qquad M(f)=\exp m(f). \] All logarithms here are natural. Roots on the unit circle, and roots equal to zero, contribute zero. The degree is allowed to vary with the polynomial. The problem asks whether \[ \forall\varepsilon>0\quad\exists\text{ monic }f\in\mathbb Z[x] \quad 0<m(f)<\varepsilon. \] Equivalently, does the set of positive logarithmic Mahler measures of monic integer polynomials have infimum zero? A negative answer would give a single constant $c>0$, independent of both degree and coefficients, such that every such polynomial has either $m(f)=0$ or $m(f)\ge c$.

The strict inequality at zero is essential: powers of $x$ and products of cyclotomic polynomials have zero measure and do not answer the question. Factorization into monic irreducible factors does not change the existence question, since Mahler measure is additive under multiplication. Thus one may equivalently seek algebraic integers of arbitrarily small positive logarithmic Mahler measure, counting all conjugates. Ward's question is the small-positive-value form of Lehmer's problem; the usual gap conjecture predicts a negative answer.

Short English statement

Can positive Mahler measures of monic integer polynomials approach zero, with no bound on their degrees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4300005 (AMR-042-0005), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ward - Six problems in Algebraic Dynamics (2006), Problem E. Original source indexed by AMR-042-0005. https://www.imath.kiev.ua/~skolyada/kevin.pdf.