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
\[M(f)\;:=\;|a_n|\prod_{i=1}^n \max\{1,|\alpha_i|\}.\]For an integer polynomial $f(x)\in\mathbb{Z}[x]$, Kronecker’s theorem characterizes the case $M(f)=1$:
\[M(f)=1 \quad\Longleftrightarrow\quad f(x)\text{ is a product of cyclotomic polynomials and }x.\]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 $L$:
\[L \;:=\; \inf\bigl\{ M(f)\;:\; f\in\mathbb{Z}[x],\ 1<M(f)\bigr\}.\]Lehmer’s original question (1933) asks whether, for every $\epsilon>0$, there exists an integer polynomial $f$ with
\[1<M(f)<1+\epsilon,\]which is equivalent to asking whether $L=1$. [BDM2007-lehmer-question]
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1.176280\ldots$ | [BDM2007] [Leh1933] | Lehmer’s example polynomial $\ell(x)=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$ has Mahler measure $M(\ell)=1.176280\ldots$, giving $L\le 1.176280\ldots$. [BDM2007-lehmer-poly] |
| $1.176280\ldots$ | [BDM2007] [Leh1933] | The value $1.176280\ldots$ (the Mahler measure of $\ell$) “remains the smallest known measure $>1$ for an integer polynomial,” i.e. it is the best currently known explicit upper bound for $L$. [BDM2007-smallest-known] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial (Kronecker) | For $f\in\mathbb{Z}[x]$, one has $M(f)\ge 1$, with $M(f)=1$ precisely in the cyclotomic/monomial case; hence $L\ge 1$. [BDM2007-kronecker] |
Additional comments and links
- Nonreciprocal case (Smyth). Smyth answered Lehmer’s question for nonreciprocal polynomials: if $f\in\mathbb{Z}[x]$ is nonreciprocal and $f(0)\neq 0$, then \(M(f)\ \ge\ M(x^3-x-1)=1.324717\ldots,\) [BDM2007-smyth-thm1] [Smy1971]
References
- [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
- [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 $M(f)=|a|\prod_{k=1}^d \max{1,|\alpha_k|}$.” - [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 $x$.” - [BDM2007-lehmer-question]
loc: Author PDF p.2, Introduction (Lehmer’s question).
quote: “In 1933, D. H. Lehmer [12] asked if for every $\epsilon > 0$ there exists a polynomial $f \in \mathbb{Z}[x]$ satisfying $1 < M(f) < 1 + \epsilon$.” - [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 $M(\ell)=1.176280\ldots$, and this value remains the smallest known measure larger than $1$ of a polynomial with integer coefficients.” - [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.” - [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 $M(f) \ge M(x^3-x-1) = 1.324717 \ldots$.”
- [BDM2007-def-M]
-
[Leh1933] Lehmer, D. H. Factorization of Certain Cyclotomic Functions. Annals of Mathematics 34(3) (1933), 461–479. DOI: 10.2307/1968172. Google Scholar.
- [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.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.