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

$$M(\alpha ):=a_d\prod _{i=1}^d\max \{1,|{\alpha }_i|\}.$$

<a href="#Vou1996-def-M">[Vou1996-def-M]</a>

Define the absolute logarithmic height $h(\alpha )$ by

$$h(\alpha ):=\frac{\log M(\alpha )}{d}.$$

<a href="#Vou1996-def-h">[Vou1996-def-h]</a>

Write

$$B(d):={\left(\frac{\log \log d}{\log d}\right)}^3(d\ge 3),$$

and consider algebraic numbers $\alpha$ that are not roots of unity. Dobrowolski proved an asymptotic lower bound of the form

$$M(\alpha )>1+(1-ϵ)B(d)\text{for }d\ge d(ϵ),$$

for each $ϵ>0$. <a href="#Vou1996-dob-asymp">[Vou1996-dob-asymp]</a>

Motivated by this asymptotic form, define the asymptotic Dobrowolski constant $C_{40b}$ to be the largest constant $c$ such that, for every $ϵ>0$, there exists $d(ϵ)$ with

$$M(\alpha )\ge 1+(c-ϵ)B(d)\text{for all non-root-of-unity }\alpha \text{ of degree }d\ge d(ϵ).$$

Known upper bounds

Known lower bounds

Additional comments and links

• $M(\alpha )$ vs. $\log M(\alpha )$. Many statements in the literature are formulated for $\log M(\alpha )$ (equivalently $dh(\alpha )$). Voutier’s inequality $$1+\log (M(\alpha ))<M(\alpha )$$ shows that any bound of the form $\log M(\alpha )>cB(d)$ immediately implies the corresponding bound $M(\alpha )>1+cB(d)$. <a href="#Vou1996-log-vs-M">[Vou1996-log-vs-M]</a>

• 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 $$M(\alpha )>1+\frac{1}{10^{4}d\log (d)}.$$ <a href="#Vou1996-bm">[Vou1996-bm]</a> <a href="#Vou1996-stew">[Vou1996-stew]</a>

References

• <a id="Vou1996"></a>[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.
• <a id="Vou1996-def-M"></a>[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 $M(\alpha )=a_d{\prod }_{i=1}^d\max (1,|{\alpha }_i|)$.”
• <a id="Vou1996-def-h"></a>[Vou1996-def-h] loc: arXiv v1 PDF p.1, Introduction (definition of $h(\alpha )$). quote: “$h(\alpha )=\frac{\log M(\alpha )}{d}$.”
• <a id="Vou1996-bm"></a>[Vou1996-bm] loc: arXiv v1 PDF p.1, Introduction (Blanksby–Montgomery). quote: “They proved that $M(\alpha )>1+\frac{1}{52d\log (6d)}$.”
• <a id="Vou1996-stew"></a>[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 $M(\alpha )>1+1/(10^{4}d\log (d))$.”
• <a id="Vou1996-dob-asymp"></a>[Vou1996-dob-asymp] loc: arXiv v1 PDF p.2, Introduction (Dobrowolski’s asymptotic bound). quote: “Dobrowolski… showed that $M(\alpha )>1+(1-ϵ){\left(\frac{\log \log d}{\log d}\right)}^3$ for $d\ge d(ϵ)$.”
• <a id="Vou1996-cs-lou"></a>[Vou1996-cs-lou] loc: arXiv v1 PDF p.2, Introduction (Cantor–Straus; Louboutin). quote: “Cantor and Straus… replace the coefficient $(1-ϵ)$ by $(2-ϵ)$. Louboutin… to $(\frac{9}{4}-ϵ)$.”
• <a id="Vou1996-log-vs-M"></a>[Vou1996-log-vs-M] loc: arXiv v1 PDF p.3, paragraph after Theorem. quote: “Notice that $1+\log (M(\alpha ))<M(\alpha )\le |\alpha {|}^d$.”

• <a id="Dob1979"></a>[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.

• <a id="CS1982"></a>[CS1982] Cantor, D.; Straus, E. G. On a conjecture of D. H. Lehmer. Acta Arithmetica 42(1) (1982), 97–100. Google Scholar.

• <a id="Lou1983"></a>[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.

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.