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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 $$\int f\lambda = -\log 2$$.

##### MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11G50 Heights
##### Keywords:
equidistribution; polynomial zeros; Mahler measure; Weil height
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##### 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
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