Asymptotic Dobrowolski constant for Lehmer’s problem
Description of constant
Let $\alpha$ be a nonzero algebraic number of degree $d$, with minimal polynomial over $\mathbb{Z}$
\[f(X)=a_d\prod_{i=1}^d (X-\alpha_i),\]where $a_d>0$ and $\alpha_1,\dots,\alpha_d$ are the conjugates of $\alpha$. Define the Mahler measure of $\alpha$ by
\[M(\alpha)\;:=\;a_d\prod_{i=1}^d \max\{1,\lvert \alpha_i\rvert\}.\]Define the absolute logarithmic height $h(\alpha)$ by
\[h(\alpha)\;:=\;\frac{\log M(\alpha)}{d}.\]Write
\[B(d)\;:=\;\left(\frac{\log\log d}{\log d}\right)^3 \qquad (d\ge 3),\]and consider algebraic numbers $\alpha$ that are not roots of unity. Dobrowolski proved an asymptotic lower bound of the form
\[M(\alpha)\;>\;1+(1-\epsilon)\,B(d)\quad \text{for }d\ge d(\epsilon),\]for each $\epsilon>0$. [Vou1996-dob-asymp]
Motivated by this asymptotic form, define the asymptotic Dobrowolski constant $C_{40b}$ to be the largest constant $c$ such that, for every $\epsilon>0$, there exists $d(\epsilon)$ with
\[M(\alpha)\;\ge\;1+(c-\epsilon)\,B(d)\quad \text{for all non-root-of-unity }\alpha\text{ of degree }d\ge d(\epsilon).\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\infty$ |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | [Dob1979] | Dobrowolski proved $M(\alpha) > 1+(1-\epsilon)B(d)$ for $d\ge d(\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 1$. [Vou1996-dob-asymp] |
| $2$ | [CS1982] | Cantor–Straus replace the coefficient $(1-\epsilon)$ by $(2-\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 2$. [Vou1996-cs-lou] |
| $9/4$ | [Lou1983] | Louboutin improves the coefficient to $(\tfrac94-\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 9/4$. [Vou1996-cs-lou] |
Additional comments and links
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$M(\alpha)$ vs. $\log M(\alpha)$. Many statements in the literature are formulated for $\log M(\alpha)$ (equivalently $d\,h(\alpha)$). Voutier’s inequality \(1+\log(M(\alpha))<M(\alpha)\) shows that any bound of the form $\log M(\alpha) > c\,B(d)$ immediately implies the corresponding bound $M(\alpha) > 1+c\,B(d)$. [Vou1996-log-vs-M]
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Earlier degree-dependent lower bounds of different shape. Before Dobrowolski’s $(\log\log d/\log d)^3$ term, Blanksby–Montgomery proved \(M(\alpha)\;>\;1+\frac{1}{52d\log(6d)},\) and Stewart proved \(M(\alpha)\;>\;1+\frac{1}{10^4d\log(d)}.\) [Vou1996-bm] [Vou1996-stew]
References
- [Vou1996] Voutier, Paul M. An effective lower bound for the height of algebraic numbers. Acta Arithmetica 74(1) (1996), 81–95. DOI: 10.4064/aa-74-1-81-95. Google Scholar. arXiv PDF.
- [Vou1996-def-M]
loc: arXiv v1 PDF p.1, Introduction (definition of $M(\alpha)$).
quote: “We shall define the Mahler measure of $\alpha$, $M(\alpha)$, by $M(\alpha)=a_d\prod_{i=1}^d\max(1,\lvert\alpha_i\rvert)$.” - [Vou1996-def-h]
loc: arXiv v1 PDF p.1, Introduction (definition of $h(\alpha)$).
quote: “$h(\alpha)=\frac{\log M(\alpha)}{d}$.” - [Vou1996-bm]
loc: arXiv v1 PDF p.1, Introduction (Blanksby–Montgomery).
quote: “They proved that $M(\alpha)>1+\frac{1}{52d\log(6d)}$.” - [Vou1996-stew]
loc: arXiv v1 PDF p.1, Introduction (Stewart).
quote: “In 1978, C.L. Stewart [18] introduced a method from transcendental number theory to prove that $M(\alpha) > 1+1/(10^4d \log(d))$.” - [Vou1996-dob-asymp]
loc: arXiv v1 PDF p.2, Introduction (Dobrowolski’s asymptotic bound).
quote: “Dobrowolski… showed that $M(\alpha)>1+(1-\epsilon)\left(\frac{\log\log d}{\log d}\right)^3$ for $d\ge d(\epsilon)$.” - [Vou1996-cs-lou]
loc: arXiv v1 PDF p.2, Introduction (Cantor–Straus; Louboutin).
quote: “Cantor and Straus… replace the coefficient $(1-\epsilon)$ by $(2-\epsilon)$. Louboutin… to $(\frac94-\epsilon)$.” - [Vou1996-log-vs-M]
loc: arXiv v1 PDF p.3, paragraph after Theorem.
quote: “Notice that $1+\log(M(\alpha))<M(\alpha)\le\lvert\alpha\rvert^d$.”
- [Vou1996-def-M]
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[Dob1979] Dobrowolski, E. On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arithmetica 34 (1979), 391–401. Google Scholar.
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[CS1982] Cantor, D.; Straus, E. G. On a conjecture of D. H. Lehmer. Acta Arithmetica 42(1) (1982), 97–100. Google Scholar.
- [Lou1983] Louboutin, R. Sur la mesure de Mahler d’un nombre algebrique. C. R. Acad. Sci. Paris Ser. I 296 (1983), 707–708. Google Scholar.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.