×

zbMATH — the first resource for mathematics

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 \(\int f\lambda = -\log 2\).

MSC:
11R04 Algebraic numbers; rings of algebraic integers
11G50 Heights
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.