Lehmer’s Mahler measure constant

Fact	Value	Source
Best lower bound with an explicit value (constant 40a)	1 Trivial (Kronecker) — For f[x], one has M(f)≥ 1, with M(f)=1 precisely in the cyclotomic/monomial case;	Tao's optimization constants
Best upper bound with an explicit value (constant 40a)	1.17628 [BDM2007] [Leh1933] — The value 1.176280 (the Mahler measure of ) “remains the smallest known measure >1 for an integer polynomial,” i.e.	Tao's optimization constants

Description of constant

Let

$f (x) = i = 0 \sum n a_{i} x^{i} = a_{n} i = 1 \prod n (x - α_{i})$

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

$M (f) := ∣ a_{n} ∣ i = 1 \prod n max {1, ∣ α_{i} ∣} .$

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

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

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

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

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

<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

Bound	Reference	Comments
$1.176280 \dots$	<a href="#BDM2007">[BDM2007]</a> <a href="#Leh1933">[Leh1933]</a>	Lehmer’s example polynomial $ℓ (x) = x^{10} + x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + x + 1$ has Mahler measure $M (ℓ) = 1.176280 \dots$ , giving $L \leq 1.176280 \dots$ . <a href="#BDM2007-lehmer-poly">[BDM2007-lehmer-poly]</a>
$1.176280 \dots$	<a href="#BDM2007">[BDM2007]</a> <a href="#Leh1933">[Leh1933]</a>	The value $1.176280 \dots$ (the Mahler measure of $ℓ$ ) “remains the smallest known measure $> 1$ for an integer polynomial,” i.e. it is the best currently known explicit upper bound for $L$ . <a href="#BDM2007-smallest-known">[BDM2007-smallest-known]</a>

Known lower bounds

Bound	Reference	Comments
$1$	Trivial (Kronecker)	For $f \in Z [x]$ , one has $M (f) \geq 1$ , with $M (f) = 1$ precisely in the cyclotomic/monomial case; hence $L \geq 1$ . <a href="#BDM2007-kronecker">[BDM2007-kronecker]</a>

Additional comments and links

Nonreciprocal case (Smyth). Smyth answered Lehmer’s question for nonreciprocal polynomials: if $f \in Z [x]$ is nonreciprocal and $f (0) \neq = 0$ , then $M (f) \geq M (x^{3} - x - 1) = 1.324717 \dots,$ <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 $f (x) = a \prod_{k = 1}^{d} (x - α_{k})$ , we have $M (f) = ∣ a ∣ \prod_{k = 1}^{d} max 1, ∣ α_{k} ∣$ .”
<a id="BDM2007-kronecker"></a>[BDM2007-kronecker] loc: Author PDF p.2, Introduction (paragraph after Equation (1.1)). quote: “For $f \in Z [x]$ , clearly $M (f) \geq 1$ , 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 $f \in Z [x]$ 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 $ℓ (x) = x^{10} + x^{9} - x^{7} - x^{6} - x^{5} - x^{4} - x^{3} + x + 1$ has $M (ℓ) = 1.176280 \dots$ , 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 $f \in Z [x]$ is nonreciprocal and $f (0) \neq = 0$ , then $M (f) \geq M (x^{3} - x - 1) = 1.324717 \dots$ .”

<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).

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Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.

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