For a nonzero polynomial P(x) = a_0∏_i=1^D(x-alpha_i) ∈ Z[x] of degree D, factored over C with leading coefficient a_0 and roots alpha_1,…,alpha_D, define its Mahler measure by M(P) = |a_0| ∏_i=1^D max big(1, |alpha_i|big), and write m(P) = log M(P). There exists an absolute constant mu > 1 such that every nonzero P(x) ∈ Z[x] satisfies at least one of the following: M(P) ≥ mu; 2. every complex root of P is either zero or a root of unity, i.e. P is an integer multiple of a product of cyclotomic polynomials and a power of the monomial x, in which case M(P) = 1. Equivalently, there is a constant c > 0 such that m(P) > c for every P(x) ∈ Z[x] having at least one root that is neither zero nor a root of unity.

Lehmer's conjecture

For a nonzero polynomial $P(x) = a_{0}\prod_{i=1}^{D}(x-\alpha_{i}) \in \mathbb{Z}[x]$ of degree $D$ , factored over $\mathbb{C}$ with leading coefficient $a_{0}$ and roots $\alpha_{1},\dots,\alpha_{D}$ , define its Mahler measure by

\mathcal{M}(P) = |a_{0}| \prod_{i=1}^{D} \max\big(1, |\alpha_{i}|\big),

and write $m(P) = \log \mathcal{M}(P)$ .

There exists an absolute constant $\mu > 1$ such that every nonzero $P(x) \in \mathbb{Z}[x]$ satisfies at least one of the following:

$\mathcal{M}(P) \geq \mu$ ; 2. every complex root of $P$ is either zero or a root of unity, i.e. $P$ is an integer multiple of a product of cyclotomic polynomials and a power of the monomial $x$ , in which case $\mathcal{M}(P) = 1$ .

Equivalently, there is a constant $c > 0$ such that $m(P) > c$ for every $P(x) \in \mathbb{Z}[x]$ having at least one root that is neither zero nor a root of unity.

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Lehmer's conjecture on the Weil height
Let $\mathrm{h}$ denote the absolute logarithmic Weil height, let $\overline{\mathbb{Q}}^*$ be the multiplicative group of nonzero algebraic numbers, and let $\mu_\infty$ be the set of roots of unity in $\overline{\mathbb{Q}}$ . Lehmer's conjecture. There exists a constant $c>0$ such that, for every $x\in\overline{\mathbb{Q}}^*\backslash\mu_\infty$ , one has
$\mathrm{h}(x)\geq\frac{c}{[\mathbb{Q}(x):\mathbb{Q}]}.$
This is a central open problem concerning lower bounds for the Weil height; it asserts a uniform inverse-degree lower bound away from roots of unity.
source: Arnaud Plessis, “Minoration de la hauteur de Weil dans un compositum de corps de rayon”, arXiv:1805.11879 (2019).
Lehmer's conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant $\mu >1$ such that every polynomial with integer coefficients $P(x)\in \mathbb {Z} [x]$ satisfies one of the following properties: The Mahler measure ${\mathcal {M}}(P(x))$ of $P(x)$ is greater than or equal to $\mu$ . $P(x)$ is an integral multiple of a product of cyclotomic polynomials or the monomial $x$ , in which case ${\mathcal {M}}(P(x))=1$ .
source: Wikipedia

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analysis unsolved-problem

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Lehmer's conjecture

Equivalent formulations 2

Lehmer's conjecture on the Weil height

Lehmer's conjecture

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