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About a question of the equidistribution of algebraic numbers. (Sur une question d’équirépartition de nombres algébriques.) (French) Zbl 1129.11046
Let \(P\) be a polynomial of degree \(d\) with integer coefficients and with zeros \(\alpha_1, \dots, \alpha_d\). Let \(\hat h(P) = \log M(P)/d\) be the Weil height of \(P\). Let \(\delta_\alpha\) be the Dirac measure with support \(\{\alpha\}\) and let \(\delta_P = (1/d)\sum_{k=1}^d \delta_{\alpha_k}\). Y. Bilu [Duke Math. J. 89, 465–476 (1997; Zbl 0918.11035)] showed that if \(P_n\) is a sequence of polynomials in \(\mathbb Z[x]\) with \(\hat h(P_n) \to 0\) then for every continuous function \(f\) on \(\mathbb P^1(\mathbb C)\), \(\int f \delta_{P_n} \to \int f \lambda\), \(\lambda\) denoting normalized Haar measure on the unit circle. J. Pineiro et al. [Geometric methods in algebra and number theory. Basel: Birkhäuser. Prog. Math. 235, 219–250 (2005; Zbl 1101.11020)], conjectured that this result should extend to \(f: z \to -\log| z-a| \). The author gives a simple example to show that this is not the case. One takes \(P_n(X) = (X^n-1)(X-2)+3\) and \(a = 2\). Then \(\hat h(P_n) = \log 5/(n+1)\), \(\int f \delta_{P_n} = - \log 3/(n+1)\) while \(\int f\lambda = -\log 2\).

11R04 Algebraic numbers; rings of algebraic integers
11G50 Heights
Full Text: DOI
[1] Bilu, Y., Limit distribution of small points on algebraic tori, Duke math. J., 89, 465-476, (1997) · Zbl 0918.11035
[2] Pineiro, J.; Szpiro, L.; Tucker, T.J., Mahler measure for dynamical systems on \(\mathbb{P}^1\) and intersection theory on a singular arithmetic surface, Prog. math., 235, 219-250, (2005) · Zbl 1101.11020
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