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PDF книги » Математика » Poulakis D., Voskos E. - On the practical solution of genus zero diophantine equations

Poulakis D., Voskos E. - On the practical solution of genus zero diophantine equations

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Название: On the practical solution of genus zero diophantine equations
Автор: Poulakis D., Voskos E.
Категория: Математика
Тип: Книга
Дата: 31.12.2008 00:30:51
Скачано: 18
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Описание: Let f(X, Y) be an absolutely irreducible polynomial with integer coefficients such that the curve С defined by the equation f(X, Y) = 0 is of genus 0. We denote by Q an algebraic closure of the field of rational numbers Q and by Q(C) the function field of С. We suppose that there are at least three discrete valuation rings of Q(C) which dominate the local rings of С at the points at infinity. Maillet (1918, 1919), using the finiteness of the integer solutions of Thue equations established in 1908, proved that the equation f(X, Y) = 0 has only finitely many integer solutions (see also, Lang, 1978, Theorem 6.1, p. 146 and 1983, Chapter 8, Section 5). The first effective upper bound for the solutions of Thue equations was obtained in 1968 by A. Baker as a consequence of his study of linear forms in the logarithms of algebraic numbers. Poulakis (1993) calculated the first effective upper bound for the integer solutions of f(X, Y) = 0 using an effective version of the Riemann—Roch theorem and an effective upper bound for the solutions of Thue equations. For other results see Bilu (1993, Theorem 5B) and Poulakis (1997, Theorem 2). Unfortunately, since the bounds obtained so far are too large, they cannot provide us with a practical method for solving the equation f(X, Y) = 0. In this paper we give a practical general method for the explicit determination of all integer solutions of a particular equation f(X, Y) = 0 satisfying the above properties. It is rested merely on the construction of a parametrization defined over Q for the points of С (if it exists) and on the practical solution of Thue equations. Since there are efficient algorithms to carry out these two tasks (see for instance Tzanakis and de Weger, 1989; Bilu and Hanrot, 1996; Sendra and Winkler, 1997), we can obtain all the integer solutions to f(X, Y) = 0 in a reasonable time. The paper is organized as follows. In Section 2 we obtain some useful results for the discussion of our method. Section 3 is devoted to the description of the algorithm for
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