## 1. Introduction

Several deep arithmetic questions are known about polynomials with integer coefficients. One of them raised by Lehmer in the 1930s asks, for a monic irreducible polynomial $P(x)=\prod _{j=1}^d(x-\alpha _j)\in \mathbb Z[x]$ , whether the quantity ${\mathrm M}(P(x))=\prod _{j=1}^d\max \{1,|\alpha _j|\}$ can be made arbitrarily close to but greater than 1. The characteristic ${\mathrm M}(P(x))$ is known as the Mahler measure [Reference Brunault and Zudilin1]; in spite of the name coined after Mahler’s work in the 1960s, many results about it are rather classical. One of them, due to Kronecker, says that ${\mathrm M}(P(x))=1$ if and only if $P(x)=x$ or the polynomial is cyclotomic, that is, all its zeros are roots of unity.

A related question, usually considered as a satellite to Lehmer’s problem, about the so-called house of a nonzero algebraic integer $\alpha $ defined through its minimal polynomial $P(x)\in \mathbb Z[x]$ as , was posed by Schinzel and Zassenhaus in the 1960s and answered only recently by Dimitrov [Reference Dimitrov2]. He proved that for any nonzero algebraic integer $\alpha $ which is not a root of unity; the latter option clearly corresponds to .

Dimitrov’s ingenious argument transforms the arithmetic problem into an analytic one. In this note we discuss the potential of Dimitrov’s approach to Lehmer’s problem.

## 2. Principal results

Consider a monic irreducible *noncyclotomic* polynomial
$P(x)=\prod _{j=1}^d(x-\alpha _j)$
in
$\mathbb Z[x]$
of degree
$d>1$
and assume that the polynomial
$\prod _{j=1}^d(x-\alpha _j^2)\in \mathbb Z[x]$
is irreducible as well. (Otherwise the Mahler measure of
$P(x)$
is bounded from below through the measures of irreducible factors of the latter polynomial.) As in [Reference Dimitrov2], Dimitrov’s cyclotomicity criterion together with Kronecker’s rationality criterion and a theorem of Pólya imply that the *hedgehog*

whose spines originate from the origin and end up at
$\alpha _j^2,\alpha _j^4$
for
$j=1,\dots ,d$
, has (logarithmic) capacity (or transfinite diameter)
$t(K)$
at least 1. Then Dubinin’s theorem [Reference Dubinin3] applies, which claims that
$t(K)\le 4^{-1/n}\max _j|\beta _j|$
(with equality attained if and only if the hedgehog *K* is rotationally symmetric), and produces the estimate for
since
$n\le 2d$
.

When dealing with Lehmer’s problem instead, one becomes interested in estimating the ‘Mahler measure of the hedgehog’, namely the quantity
$\prod _{j=1}^n\max \{1,|\beta _j|\}$
, because any nontrivial (bounded away from 1) absolute estimate for it would imply a nontrivial estimate for the Mahler measure of
$P(x)$
. In this setting, Dubinin’s theorem only implies the estimate
$\prod _{j=1}^n\max \{1,|\beta _j|\}\ge 4^{1/n}$
for a hedgehog of capacity at least 1, which depends on *n*. The Mahler measure of the rotationally symmetric hedgehog on *n* spines, which is optimal in Dubinin’s result, is equal to 4 (thus, independent of *n*), which certainly loses out to the Mahler measure
$1.91445008\dots $
of the ‘Lehmer hedgehog’ attached to the polynomial
$x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$
but also to the measure
$3.07959562\dots $
of the hedgehog constructed on Smyth’s polynomial
$x^3-x-1$
. The following question arises in a natural way.

Question 1. What is the minimum of $\prod _{j=1}^n\max \{1,|\beta _j|\}$ taken over all hedgehogs $K=K(\beta _1,\dots ,\beta _n)$ of capacity at least $1$ ?

Notice that answering this question for hedgehogs of capacity exactly $1$ is sufficient, since the capacity satisfies $t(K_1)\le t(K_2)$ for any compact sets $K_1\subset K_2$ in $\mathbb C$ .

In order to approach Question 1 we use a different construction of hedgehogs outlined in Eremenko’s post on the question in [Reference Lev5] with details set out in [Reference Schmidt6]. Any hedgehog
$K=K(\beta _1,\dots ,\beta _n)$
of capacity *precisely*
$1$
is in a bijective correspondence (up to rotation!) with the set of points
$z_1,\dots ,z_n$
on the unit circle with prescribed *positive* real weights
$r_1,\dots ,r_n$
satisfying
$r_1+\dots +r_n=1$
. Namely, the mapping

is a Riemann mapping of the complement of the closed unit disk to the complement $\hat {\mathbb C}\setminus K$ of hedgehog. It is not easy to write down the corresponding $\beta _j$ explicitly, but for their absolute values we get

where we conventionally take
$z_0=z_n$
and understand
$[z_{j-1},z_j]$
as *arcs* of the unit circle. This means that if
$C\ge 1$
is the minimum of

taken over all *n* and all possible weighted configurations
$z_1,\dots ,z_n$
, then
$C^2$
is the minimum in Question 1.

Furthermore, in the spirit of [Reference Konyagin and Lev4] observe that from continuity considerations it suffices to compute the required minimum *C* for *rational* positive weights
$r_1,\dots ,r_n$
. Assuming the latter and writing
$r_j=a_j/m$
for positive integers
$a_1,\dots ,a_n$
and
$m=a_1+\dots +a_n$
, we look for the *m*th root of the minimum of

where $z_1',z_2',\dots ,z_m'$ is the multi-set

with prescribed weights all equal to 1. This means that it is enough to compute the minimum for the case of equal weights, $r_1=\dots =r_n=1/n$ , and we may give the following alternative formulation of Question 1.

Question 2. What is the minimum $C_n$ of

taken over all configurations of points $z_1,\dots ,z_n$ on the unit circle $|z|=1$ ? The points are not required to be distinct and $[z_{j-1},z_j]$ is understood as the corresponding arc of the circle, $z_0$ is identified with $z_n$ .

Though there is no explicit requirement on the order of precedence, the minimum corresponds to the successive locations of $z_1,\dots ,z_n$ on the circle.

A comparison with Dubinin’s result suggests that good candidates for the minima in Question 2 may originate from configurations in which all factors in the defining product but one are equal to $1$ . In our answer to the question we show that this is essentially the case by computing the related minima $C_n^*$ explicitly.

Theorem 3. For the quantity $C_n$ we have the inequality $C_n\le C_n^*$ , where $C_n^*=(T_n(2^{1/n}))^{1/n}$ and

denotes the *n*th Chebyshev polynomial of the first kind.

Theorem 4. For the quantity $C_n^*$ in Theorem 3 we have the asymptotic expansion

in terms of $\nu =\sqrt {(\log 4)/n}$ , as $n\to \infty $ . In particular, $(C_n^*)^{\sqrt n}\to e^{\sqrt {\log 4}}$ and $C_n^*\to 1$ as $n\to \infty $ .

Thus, our results imply that the minimum in Question 1 is equal to $1$ , meaning that an analogue of Lehmer’s problem in an analytic setting is trivial. This has no consequences for Lehmer’s problem itself, as we are not aware of a recipe to cook up polynomials in $\mathbb Z[x]$ from optimal (or near optimal) configurations of $z_1,\dots ,z_n$ on the unit circle.

## 3. Proofs

## Proof of Theorem 3

We look for a configuration of the points $z_1,\dots ,z_n$ on the unit circle such that the maximum of $|Q(z)|$ , where $Q(z)=(z-z_1)\dotsb (z-z_n)$ , on all the arcs $[z_{j-1},z_j]$ but one is equal to $1$ :

At the same time, the *k*th Chebyshev polynomial
$T_k(x)=2^{k-1}x^k+\dotsb $
is known to satisfy
$|T_k(x)|\le 1$
on the interval
$-1\le x\le 1$
, with *all* the extrema on the interval being either
$-1$
or
$1$
. Note that
$T_k(x)$
has *k* distinct real zeros on the open interval
$-1<x<1$
and satisfies
$T_k(1)=(-1)^kT_k(-1)=1$
. Therefore, for
$n=2k$
even,

we get a *monic* polynomial of degree *n* with the desired properties; its zeros
$z_1,\dots ,z_n$
ordered in pairs, so that
$z_{n-j}=\overline z_j=z_j^{-1}$
for
$j=1,\dots ,k$
, correspond to the real zeros
$2^{1/k}((z_j+z_j^{-1})/2-1)+1$
of the polynomial
$T_k(x)$
on the interval
$-1<x<1$
. Then

where the duplication formula $T_k(2x^2-1)=T_{2k}(x)$ was applied.

The duplication formula in fact allows one to write the very same polynomial $Q(z)$ in the form

and this formula gives the desired polynomial, monic and of degree *n*, for *n* of any parity. If we set
$k=\lfloor (n+1)/2\rfloor $
, the zeros
$z_1,\dots ,z_n$
of
$Q(z)$
pair as before, that is,
$z_{n-j}=\overline z_j=z_j^{-1}$
for
$j=1,\dots ,k$
, with the two zeros merging into one,
$z_{(n+1)/2}=1$
for
$j=k$
when *n* is odd, so that
$2^{1/n - 1} \sqrt {2- \smash [b]{(z_j+z_j^{-1})}}$
for
$j=1,\dots ,k$
are precisely the *k* real zeros of the polynomial
$T_n(x)$
on the interval
$0\le x<1$
. This leads to the estimate

for both even and odd values of *n*.

Finally, we remark that the *uniqueness* of
$Q(z)$
, up to rotation, follows from the extremal properties of the Chebyshev polynomials.

## Proof of Theorem 4

For this part we cast the Chebyshev polynomial $T_n(x)$ in the form

leading to

in the notation $\nu =\sqrt {(\log 4)/n}$ . Since

we conclude that the term $(1-\sqrt {1-e^{-\nu ^2}})^n=O(\varepsilon ^n)$ for any choice of positive $\varepsilon <1$ , hence

and the required asymptotics follows.

## 4. Speculations

Dimitrov’s estimate $t(K)\ge 1$ for the capacity of the hedgehog $K=K(\beta _1,\dots ,\beta _n)$ assigned to a polynomial in $\mathbb Z[x]$ is not necessarily sharp, and one would rather expect to have $t(K)\ge t$ for some $t>1$ . By replacing the polynomial in the proof of Theorem 3 with

and assuming (or, better, believing!) that the corresponding minimum in Question 2 is indeed attained in the case when all but one of the factors are equal to 1, we conclude that the minimum is equal to $(T_n(2^{1/n}t))^{\! 1/n}$ . The asymptotics of the Chebyshev polynomials then converts this result into the answer

to the related version of Question 1. This is slightly better, when
$t>1$
, than the trivial estimate of the infimum by *t* from below.

In another direction, one may try to associate hedgehogs *K* to polynomials in a different (more involved!) way, to achieve some divisibility properties for the Hankel determinants
$A_k$
that appear in the estimation
$t(K)\ge \limsup _{k\to \infty }|A_k|^{1/k^2}$
of the capacity on the basis of Pólya’s theorem. Such an approach has the potential to lead to some partial (‘Dobrowolski-type’) resolutions of Lehmer’s problem. Notice, however, that the bound for
$t(K)$
in Pólya’s theorem is not sharp: numerically, the Hankel determinants
$A_k=\det _{0\le i,j<k}(a_{i+j})$
constructed on (Dimitrov’s) irrational series

for Smyth’s polynomial $x^3-x-1=(x-\alpha _1)(x-\alpha _2)(x-\alpha _3)$ satisfy $|A_k|\le C^k$ for some $C<2.5$ and all $k\le 150$ , so that it is likely that $\limsup _{k\to \infty }|A_k|^{1/k^2}=1$ in this case.

## Acknowledgement

The third author thanks Yuri Bilu and Laurent Habsieger for inspirational conversations on the Lehmer and Schinzel–Zassenhaus problems.