Frisch, Sophie; Lettl, Günter Polynomial parametrization of the solutions of Diophantine equations of genus 0. (English) Zbl 1214.11043 Funct. Approximatio, Comment. Math. 39, Part 2, 205-209 (2008). 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 \[ L_f\setminus L_f^{bad} = \{(g_1(x),g_2(x),g_3(x))\mid x \in {\mathbb Z}^m\}. \] Reviewer: Dimitros Poulakis (Thessaloniki) 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 Keywords:Diophantine equation, curves of genus 0; integer valued polynomials, polynomial parametrization PDFBibTeX XMLCite \textit{S. Frisch} and \textit{G. Lettl}, Funct. Approximatio, Comment. Math. 39, Part 2, 205--209 (2008; Zbl 1214.11043) Full Text: DOI arXiv Euclid References: [1] S. Frisch, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra 36 (2008), 1110–1114. · Zbl 1209.11038 · doi:10.1080/00927870701776938 [2] S. Frisch, L. Vaserstein, Parametrization of Pythagorean triples by a single triple of polynomials, J. Pure Appl. Algebra 212 (2008), 271–274. · Zbl 1215.11025 · doi:10.1016/j.jpaa.2007.05.019 [3] E. Kunz, Introduction to Plane Algebraic Curves, Birkhäuser, 2005. · Zbl 1078.14041 [4] D. Poulakis, E. Voskos, Solving genus zero Diophantine equations with at most two infinite valuations, J. Symbolic Computation 33 (2002), 479–491. · Zbl 0998.11014 · doi:10.1006/jsco.2001.0515 [5] R.,J. Walker, Algebraic Curves, Springer, 1978. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.