Hedgehogs in Lehmer's problem

Motivated by a famous question of Lehmer about the Mahler measure we study and solve its analytic analogue.


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 irre- , whether the quantity M(P (x)) = d j=1 max{1, |α j |} can be made arbitrary close to but larger than 1. The characteristic M(P (x)) is known as the Mahler measure [1]; in spite of the name coined after Mahler's works in the 1960s, many results about it are rather classical. One of them, due to Kronecker, says that 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 non-zero algebraic integer α defined through its minimal polynomial P (x) ∈ Z[x] as α = max j |α j |, was asked by Schinzel and Zassenhaus in the 1960s and answered only recently by Dimitrov [2]. He proved that α ≥ 2 1/(4d) for any non-zero algebraic integer α which is not a root of unity; the latter option clearly corresponds to α = 1.
Dimitrov's ingenious argument transforms the arithmetic problem into an analytic one. In this note we discuss potentials of Dimitrov's approach to Lehmer's problem.

Principal results
Consider a monic irreducible non-cyclotomic polynomial P ( 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 [2], Dimitrov's cyclotomicity criterion together with Kronecker's rationality criterion Date: 31 May 2021. The work of the second author is supported by NWO grant OCENW.KLEIN.006. 1 and a theorem of Pólya imply that the hedgehog whose spines originate from the origin and end up at α 2 j , α 4 j for j = 1, . . . , d, has (logarithmic) capacity (aka transfinite diameter) t(K) at least 1. Then Dubinin's theorem [3] applies, which claims that t(K) ≤ 4 −1/n max j |β j | (with the equality attained if and only if the hedgehog K is rotationally symmetric), and produces the estimate for α 1 = max j |β j | 1/4 since n ≤ 2d.
When dealing with Lehmer's problem instead, one becomes interested in estimating the 'Mahler measure of hedgehog', namely the quantity n j=1 max{1, |β j |}, because any non-trivial (bounded away from 1) absolute estimate for it would imply a non-trivial estimate for the Mahler measure of P (x). In this setting, Dubinin's theorem only implies the estimate n j=1 max{1, |β j |} ≥ 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 to the Mahler measure 1.91445008 . . . 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 . . . of 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 n j=1 max{1, |β j |} taken over all hedgehogs K = K(β 1 , . . . , β 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 ) ≤ t(K 2 ) for any compacts K 1 ⊂ K 2 in C.
In order to approach Question 1 we use a different construction of hedgehogs outlined in the post of Eremenko to the question in [5] with details exposed in [6]. Any hedgehog K = K(β 1 , . . . , β n ) of capacity precisely 1 is in a bijective correspondence (up to rotation!) with the set of points z 1 , . . . , z n on the unit circle with prescribed positive real weights r 1 , . . . , r n satisfying r 1 + · · · + r n = 1. Namely, the mapping is a Riemann mapping of the complement of the closed unit disk to the complement C \ K of hedgehog. It is not easy to write down the corresponding β j explicitly but for their absolute values we get where we take conventionally z 0 = z n and understand [z j−1 , z j ] as arcs of the unit circle. It means that if C ≥ 1 is the minimum of n j=1 max 1, max |z − z k | r k taken over all n and all possible weighted configurations z 1 , . . . , z n , then C 2 is the minimum in Question 1.
Furthermore, in the spirit of [4] observe that from the continuity considerations it suffices to compute the required minimum C for rational positive weights r 1 , . . . , r n . Assuming the latter and writing r j = a j /m for positive integers a 1 , . . . , a n and m = a 1 + · · · + a n , we are for the mth root of the minimum of n j=1 max 1, max 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 = · · · = r n = 1/n, and we may give the following alternative formulation of Question 1. Though there is no explicit requirement on the order of precedence, the minimum corresponds to the successive location of z 1 , . . . , 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 1. For the quantity C n we have the inequality C n ≤ C * n , where C * n = T n (2 1/n ) 1/n and denotes the nth Chebyshev polynomial of the first kind.
Theorem 2. For the quantity C * n in Theorem 1 we have the asymptotic expansion in terms of ν = (log 4)/n, as n → ∞. In particular, (C * n ) √ n → e √ log 4 and C * n → 1 as n → ∞.
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 brings no consequences to Lehmer's problem itself, as we are not aware of a recipe to cook up polynomials in Z[x] from optimal (or near optimal) configurations of z 1 , . . . , z n on the unit circle.

Proofs
Proof of Theorem 1. We look for a configuration of the points z 1 , . . . , z n on the unit circle such that the maximum of |Q(z)|, where Q(z) = (z − z 1 ) · · · (z − z n ), on all the arcs [z j−1 , z j ] but one is equal to 1: At the same time, the kth Chebyshev polynomial T k (x) = 2 k−1 x k + · · · is known to satisfy |T k (x)| ≤ 1 on the interval −1 ≤ x ≤ 1, with all the extrema on the interval to be either −1 or 1. Note that T k (x) has k distinct real zeroes on the open interval −1 < x < 1 and satisfies T k (1) = (−1) k T k (−1) = 1. Therefore, setting for n = 2k even, we get a monic polynomial of degree n with the desired properties; its zeroes z 1 , . . . , z n ordered in pairs, z n−j = z j = z −1 j for j = 1, . . . , k, correspond to the real zeroes 2 1/k (z j + z −1 j )/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 Q(z) = ±(−z) n/2 T n 2 1/n−1 2 − (z + z −1 ) , and this formula gives the wanted polynomial, monic and of degree n, for n of any parity. If we set k = ⌊(n + 1)/2⌋, the zeroes z 1 , . . . , z n of Q(z) pair as before, z n−j = z j = z −1 j for j = 1, . . . , k, with the two zeroes merging into one, z (n+1)/2 = 1 for j = k when n is odd, so that 2 1/n−1 2 − (z j + z −1 j ) for j = 1, . . . , k are precisely the k real zeroes of the polynomial T n (x) on the interval 0 ≤ x < 1. This leads to the estimate max Finally, we remark that the uniqueness of Q(z), up to rotation, follows from the extremal properties of the Chebyshev polynomials.
Proof of Theorem 2. For this part we cast the Chebyshev polynomial T n (x) in the form in the notation ν = (log 4)/n. Since we conclude that the term (1− √ 1 − e −ν 2 ) n = O(ε n ) for any choice of positive ε < 1, hence and the required asymptotics follows.

Speculations
Dimitrov's estimate t(K) ≥ 1 for the capacity of the hedgehog K = K(β 1 , . . . , β n ) assigned to a polynomial in Z[x] is not necessarily sharp, and one would rather expect to have t(K) ≥ t for some t > 1. By replacing the polynomial in the proof of Theorem 1 with Q(z) = ±(−z) n/2 T n 2 1/n−1 t 2 − (z + z −1 ) and assuming (or, better, believing!) that the corresponding minimum in Question 2 is indeed attained in the case when all but one 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 inf n=1,2,...
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 estimation t(K) ≥ lim sup k→∞ |A k | 1/k 2 of the capacity on the basis of Pólya's theorem. Such an approach has 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≤i,j<k (a i+j ) constructed on (Dimitov's) irrational series for Smyth's polynomial x 3 − x − 1 = (x − α 1 )(x − α 2 )(x − α 3 ) satisfy |A k | ≤ C k for some C < 2.5 and all k ≤ 150, so that it is likely that lim sup k→∞ |A k | 1/k 2 = 1 in this case.