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Polynomial parametrization of the solutions of Diophantine equations of genus 0. (English) Zbl 1214.11043

Let \(f \in {\mathbb Z}[X,Y,Z]\) be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients and \(C_f\) the plane projective curve defined by the equation \(f = 0\). Suppose that the function field \( {\mathbb Q}(C_f)\) of \(C_f\) is isomorphic to \( {\mathbb Q}(T)\). We denote by \(L_f\) the set of \((x,y,z) \in {\mathbb Z}^3\) satisfying \(f(x,y,z) = 0\) and by \(L_f^{bad}\) the subset of \(L_f\) which contains the nonsingular points of \(L_f\). In this paper it is proved that there exist polynomials \(g_1,g_2,g_3 \in {\mathbb Z}[X_1,\ldots,X_m]\), for some \(m\), such that \(g_i( {\mathbb Z}^m) \subset {\mathbb Z}\) and \[ L_f\setminus L_f^{bad} = \{(g_1(x),g_2(x),g_3(x))\mid x \in {\mathbb Z}^m\}. \]

MSC:

11D85 Representation problems
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11D41 Higher degree equations; Fermat’s equation
14H05 Algebraic functions and function fields in algebraic geometry
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References:

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