Lehmer’s Mahler measure constant

Description of constant

Let

$$f(x)=\sum _{i=0}^n{a}_i{x}^i=a_n\prod _{i=1}^n(x-{\alpha }_i)$$

be a polynomial with complex coefficients. The Mahler measure of $f$ is

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

<a href="#BDM2007-def-M">[BDM2007-def-M]</a>

For an integer polynomial $f(x)\in \mathbb{Z}[x]$, Kronecker’s theorem characterizes the case $M(f)=1$:

$$M(f)=1⟺f(x)\text{ is a product of cyclotomic polynomials and }x.$$

<a href="#BDM2007-kronecker">[BDM2007-kronecker]</a>

Motivated by Lehmer’s question, define Lehmer’s Mahler measure constant $C_{40a}$ to be the infimum of Mahler measures strictly larger than $1$ among integer polynomials, and denote it by $L$:

$$L:=\inf \{M(f):f\in \mathbb{Z}[x],1<M(f)\}.$$

<a href="#BDM2007-lehmer-question">[BDM2007-lehmer-question]</a>

Lehmer’s original question (1933) asks whether, for every $ϵ>0$, there exists an integer polynomial $f$ with

$$1<M(f)<1+ϵ,$$

which is equivalent to asking whether $L=1$. <a href="#BDM2007-lehmer-question">[BDM2007-lehmer-question]</a>

Known upper bounds

Known lower bounds

Additional comments and links

• Nonreciprocal case (Smyth). Smyth answered Lehmer’s question for nonreciprocal polynomials: if $f\in \mathbb{Z}[x]$ is nonreciprocal and $f(0)\ne 0$, then $$M(f)\ge M(x^3-x-1)=1.324717\dots ,$$ <a href="#BDM2007-smyth-thm1">[BDM2007-smyth-thm1]</a> <a href="#Smy1971">[Smy1971]</a>

References

• <a id="BDM2007"></a>[BDM2007] Borwein, Peter; Dobrowolski, Edward; Mossinghoff, Michael J. Lehmer’s problem for polynomials with odd coefficients. Annals of Mathematics 166(2) (2007), 347–366. DOI: 10.4007/annals.2007.166.347. Google Scholar. Author PDF
• <a id="BDM2007-def-M"></a>[BDM2007-def-M] loc: Author PDF p.1, Equation (1.1). quote: “Mahler’s measure of a polynomial $f$, denoted $M(f)$, is defined as the product of the absolute values of those roots of $f$ that lie outside the unit disk, multiplied by the absolute value of the leading coefficient. Writing $f(x)=a{\prod }_{k=1}^d(x-{\alpha }_k)$, we have $$\begin{gathered} M(f)=|a|{\prod }_{k=1}^d\max \\ 1,|{\alpha }_k| \end{gathered}$$.”
• <a id="BDM2007-kronecker"></a>[BDM2007-kronecker] loc: Author PDF p.2, Introduction (paragraph after Equation (1.1)). quote: “For $f\in \mathbb{Z}[x]$, clearly $M(f)\ge 1$, and by a classical theorem of Kronecker, $M(f)=1$ precisely when $f(x)$ is a product of cyclotomic polynomials and the monomial $x$.”
• <a id="BDM2007-lehmer-question"></a>[BDM2007-lehmer-question] loc: Author PDF p.2, Introduction (Lehmer’s question). quote: “In 1933, D. H. Lehmer [12] asked if for every $ϵ>0$ there exists a polynomial $f\in \mathbb{Z}[x]$ satisfying $1<M(f)<1+ϵ$.”
• <a id="BDM2007-lehmer-poly"></a>[BDM2007-lehmer-poly] loc: Author PDF p.2, Introduction (Lehmer’s example). quote: “Lehmer noted that the polynomial $\ell (x)=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$ has $M(\ell )=1.176280\dots$, and this value remains the smallest known measure larger than $1$ of a polynomial with integer coefficients.”
• <a id="BDM2007-smallest-known"></a>[BDM2007-smallest-known] loc: Author PDF p.2, Introduction (after Lehmer’s example). quote: “and this value remains the smallest known measure larger than $1$ of a polynomial with integer coefficients.”
• <a id="BDM2007-smyth-thm1"></a>[BDM2007-smyth-thm1] loc: Author PDF p.2, Introduction (Smyth’s result). quote: “Smyth [22] showed that if $f\in \mathbb{Z}[x]$ is nonreciprocal and $f(0)\ne 0$, then $M(f)\ge M(x^3-x-1)=1.324717\dots$.”

• <a id="Leh1933"></a>[Leh1933] Lehmer, D. H. Factorization of Certain Cyclotomic Functions. Annals of Mathematics 34(3) (1933), 461–479. DOI: 10.2307/1968172. Google Scholar.

• <a id="Smy1971"></a>[Smy1971] Smyth, C. J. On the product of the conjugates outside the unit circle of an algebraic integer. Bulletin of the London Mathematical Society 3(2) (1971), 169–175. DOI: 10.1112/blms/3.2.169. 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.