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Hedgehogs in Lehmer’s problem. (English) Zbl 1494.11089

Motivated by a recent V. Dimitrov’s proof of Schinzel and Zassenhaus conjecture [“A proof of the Schinzel-Zassenhaus conjecture on polynomials”, Preprint, arXiv:1912.12545] the authors study the minimum of \(\prod_{j=1}^n \max(1,|\beta_j|)\) taken over all hedgehogs \[ K =K(\beta_1,\dots,\beta_n) = \cup_{k=1}^n [0,\beta_k] \] of capacity at least 1. The problem is transferred to the study of the minimum \(C_n\) of \[ \prod_{j=1}^n \max \bigg(1,\max_{z \in [z_{j-1},z_j]} \prod_{k=1}^n |z-z_k|\bigg)^{1/n}, \] where the maximum is taken over 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 \(z_n\).
They show that \(C_n \leq T_n(2^{1/n})^{1/n}\), where \[ T_n(x)=\sum_{k=0}^{[n/2]} \binom{n}{2k} (x^2-1)^k x^{n-2k} \] is the \(n\)th Chebyshev polynomial of the first kind. They also derive the asymptotic expansion (as \(n \to \infty\)) \[ T_n(2^{1/n})^{1/n}=1+\nu-\frac{\nu^3}{4}+\frac{5\nu^5}{96}-\frac{\nu^7}{128}+O(\nu^9), \] where \(\nu=\sqrt{(\log 4)/n}\).

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
30E10 Approximation in the complex plane
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

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References:

[1] Brunault, F. and Zudilin, W., Many Variations of Mahler Measures: A Lasting Symphony, Australian Mathematical Society Lecture Series, 28 (Cambridge University Press, Cambridge, 2020). · Zbl 1476.11002
[2] Dimitrov, V., ‘A proof of the Schinzel-Zassenhaus conjecture on polynomials’, Preprint, 2019.
[3] Dubinin, V. N., ‘On the change in harmonic measure under symmetrization’, Mat. Sb.52(1) (1985), 267-273. · Zbl 0571.30025
[4] Konyagin, S. V. and Lev, V. F., ‘On the maximum value of polynomials with given degree and number of roots’, Chebyshevskiǐ Sb.3(2(4)) (2002), 165-170. · Zbl 1102.30004
[5] Lev, V. F., ‘The maximum of a polynomial on the unit circle’, MathOverflow, 2011, https://mathoverflow.net/q/64099.
[6] Schmidt, H., ‘Explicit Riemann mappings for hedgehogs’, Preprint, 2020.
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