van Hoeij, Mark; Cremona, John Solving conics over functions fields. (English) Zbl 1129.11053 J. Théor. Nombres Bordx. 18, No. 3, 595-606 (2006). Let \(F\) be a field of characteristic other than 2, and \(a,b,c\in F[t] \backslash \{0\}\) with \(abc\) square free. Let \(u\) be a new variable and set \(f_a=bu^2+c\), \(f_b=cu^2+a\), and \(f_c=au^2+b\). For an irreducible polynomial \(p\in F[t]\) we denote \(L_p=F[t]/(p)\). Further, if \(f\in F[t][u]\), then we denote by \(f\) mod \(p\) the image of \(f\) in \(L_p [u]\). Finally, we denote by supp\((a)\) the set of all monic irreducible \(p\in F[t]\) that divide \(a\). A solubility certificate for the equation \[ aX^2+ bY^2+cZ^2=0\tag{1} \] is a list containing the following:\(\bullet\) For every \(p\in\text{supp}(a)\), a root of \(f_a\bmod p\) in \(L_p\);\(\bullet\) For every \(p\in\text{supp}(b)\), a root of \(f_b\) mod \(p\) in \(L_p\);\(\bullet\) For every \(p\in\text{supp}(c)\), a root of \(f_c\) mod \(p\) in \(L_p\);\(\bullet\) If \(\deg(a)\equiv\deg(b)=\equiv\deg(c) \pmod 2\) and \(abc\) has no root in \(F\), then either a solution or a solubility certificate for the equation \[ l_aX^2+l_bY^2+l_xZ^2=0, \] over \(F(t)\), where \(l_a,l_b,l_c\) are the leading coefficients of \(a,b\), and \(c\), respectively. In the paper under review, the authors prove that equation (1) has a solution in the projective plane \(P^2 (F(t))\) if and only if a solubility certificate exists; in this case they give a simple algorithm which determines a such solution. Moreover, they describe an algorithm that reduces an equation \(aX^2+bY^2+cZ^2=0\), where \(a,b,c\in F(t)^*\), to an equation of the same form with \(a,b,c\in F[t]\setminus\{0\}\) and \(abc\) square free. Reviewer: D. Poulakis (Thessaloniki) Cited in 7 Documents MSC: 11R58 Arithmetic theory of algebraic function fields 11Y50 Computer solution of Diophantine equations 68W30 Symbolic computation and algebraic computation Keywords:conic; solubility certificate; reduced term Software:Magma; Maple PDFBibTeX XMLCite \textit{M. van Hoeij} and \textit{J. Cremona}, J. Théor. Nombres Bordx. 18, No. 3, 595--606 (2006; Zbl 1129.11053) Full Text: DOI Numdam EuDML EMIS References: [1] T. Cochrane, P. Mitchell, Small solutions of the Legendre equation. J. Number Theory 70 (1998), no. 1, pp. 62-66. · Zbl 0908.11012 [2] J. Cremona, D. Rusin, Efficient solution of rational conics. Math. Comp. 72 (2003), no. 243, pp. 1417-1441. · Zbl 1022.11031 [3] C. F. Gauss, Disquisitiones Arithmeticae. Springer-Verlag, 1986. · Zbl 0585.10001 [4] A. K. Lenstra, H. W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), no. 4, pp. 515-534. · Zbl 0488.12001 [5] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput., 24 (1997), 235-265. Computational algebra and number theory (London, 1993). See also . · Zbl 0898.68039 [6] M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, Maple 6 Programming Guide. Waterloo Maple Inc. (Waterloo, Canada, 2000). [7] M. Reid, Chapters on Algebraic Surfaces, Chapter C: Guide to the classification of surfaces. In J. Kollár (Ed.), IAS/Park City lecture notes series 3 (1993), AMS, Providence R.I., 1997, 1-154. See also . · Zbl 0910.14016 [8] J. Schicho, Rational parametrization of real algebraic surfaces. Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (Rostock), ACM, New York, 1998, 302-308. · Zbl 0939.14034 [9] J. Schicho, Proper parametrization of surfaces with a rational pencil. Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (St. Andrews), ACM, New York, 2000, 292-300. · Zbl 1326.68369 [10] D. Simon, Solving quadratic equations using reduced unimodular quadratic forms. Math. Comp. 74 (2005), no. 251, pp. 1531-1543. · Zbl 1078.11072 [11] C. van de Woestijne, Surface Parametrisation without Diagonalisation. Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (Genoa), ACM, New York, 2006, 340-344. · Zbl 1356.65061 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.