Explicit lower bounds for linear forms in two logarithms

Nicolas Gouillon

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Let f(z), $z = r e^{i θ}$ , be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values $I_{δ} (r)$ and the iterated mean values $N_{δ, k} (r)$ of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).

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In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation $\begin{matrix} Φ_{n} (x, y) & = x^{3} + (n^{8} + 2 n^{6} - 3 n^{5} + 3 n^{4} - 4 n^{3} + 5 n^{2} - 3 n + 3) x^{2} y \\ - (n^{3} - 2) n^{2} x y^{2} - y^{3} = \pm 1, \end{matrix}$ for $n \geq 0$ .

A constant in pluripotential theory

Zbigniew Błocki (1992)

Annales Polonici Mathematici

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