## 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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