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Computing the Mahler measure of a polynomial using Graeffe iterations. (Calcul numérique de la mesure de Mahler d’un polynôme par itérations de Graeffe.) (French) Zbl 0834.30006

Let \(P(z)= a_0 z^n+\cdots+ a_n= a_0 \prod_i (z- \alpha_i)\) be a polynomial with complex coefficients and let \(P_m(z)= a^{2^m}_0 \prod_i (z\alpha^{2^m}_i)\) be the polynomial obtained by \(m\) iterations of Graeffe’s method. Let \(M(P)= |a_0|\prod_i \max(1, |\alpha_i |)\) denote the Mahler measure of \(P\) and let \(|P|_2\) and \([P]_2\) denote the \(L_2\) and Bombieri norms of \(P\).
The reviewer pointed out that \(|P_m|^{1/2^m}\) converges to \(M(P)\) with an error which is estimable by use of inequalities of Mahler [the reviewer, Math. Comput. 35, 1361-1377 (1980; Zbl 0447.12002)], but that the convergence was slow if \(P\) has roots near the unit circle. The authors quantify this statement and a similar statement for the sequence \([P_m]^{1/2^m}_2\) giving the precise rate of convergence if all roots of \(P\) lie outside the annulus \(1/r< |z|< r\) for some \(r> 1\).

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

Zbl 0447.12002
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