Proper real reparametrization of rational ruled surfaces

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Let K subset of R be a computable field. We present an algorithm to decide whether a proper rational parametrization of a ruled surface, with coefficients in K((i), can be properly reparametrized over a real (i.e. embedded in R) finite field extension of K. Moreover, in the affirmative case, the algorithm provides a proper parametrization with coefficients in a real extension of K of degree at most 2.
Andradas, C., Recio, T., Sendra, J.R., 1999. Base field restriction techniques for parametric curves. In: Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC). ACM, New York, pp. 17–22 (electronic). Andradas, C., Recio, T., Sendra, J.R., Tabera, L.F., 2009. On the simplification of the coefficients of a parametrization. J. Symbolic Comput. 44 (2), 192–210. Buchberger, B., Collins, G.E., Loos, R., Albrecht, R. (Eds.), 1983. Computer Algebra. Second edition. Springer-Verlag, Vienna. Symbolic and algebraic computation. Dohm, M., 2009. Implicitization of rational ruled surfaces with μ-bases. J. Symbolic Comput. 44 (5), 479–489. Landsmann, G., Schicho, J., Winkler, F., Hillgarter, E., 2000. Symbolic parametrization of pipe and canal surfaces. In: ISSAC 2000. ACM, pp. 202–208. Langemyr, L., McCallum, S., 1989. The computation of polynomial greatest common divisors over an algebraic number field. J. Symbolic Comput. 8 (5), 429–448. Li, J., Shen, L.-Y., Gao, X.-S., 2008. Proper reparametrization of rational ruled surface. J. Comput. Sci. Tech. 23 (2), 290–297. Pérez-Díaz, S., 2006. On the problem of proper reparametrization for rational curves and surfaces. Comput. Aided Geom. Design 23 (4), 307–323. Peternell, M., Pottmann, H., 1997. Computing rational parametrizations of canal surfaces. In: Parametric Algebraic Curves and Applications. Albuquerque, NM, 1995, J. Symbolic Comput. 23 (2–3), 255–266. Recio, T., Sendra, J.R., 1997. Real reparametrizations of real curves. In: Parametric Algebraic Curves and Applications. Albuquerque, NM, 1995, J. Symbolic Comput. 23 (2–3), 241–254. Recio, T., Sendra, J.R., Tabera, L.F., Villarino, C., 2010. Generalizing circles over algebraic extensions. Math. Comput. 79 (270), 1067–1089. Schicho, J., 1998a. Rational parameterization of real algebraic surfaces. In: Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation (Rostock).ACM, New York,pp.302–308 (electronic). Schicho, J., 1998b. Rational parametrization of surfaces. J. Symbolic Comput. 26 (1), 1–29. Schicho, J., 2000a. Proper parametrization of real tubular surfaces. J. Symbolic Comput. 30 (5), 583–593. Schicho, J., 2000b. Proper parametrization of surfaces with a rational pencil. In: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (St. Andrews). ACM, New York, pp. 292–300 (electronic). Sederberg, T.W., Snively, J.P., 1987. Parametrization of cubic algebraic surfaces. In: The Mathematics of Surfaces, II. Cardiff, 1986. In: Inst. Math. Appl. Conf. Ser. New Ser., vol. 11. Oxford Univ. Press, New York, pp. 299–319. Sendra, J.R., Winkler, F., Pérez-Díaz, S., 2008. Rational Algebraic Curves: A Computer Algebra Approach. Algorithms and Computation in Mathematics, vol. 22. Springer, Berlin. Shafarevich, I.R., 1994. Varieties in projective space. In: Basic Algebraic Geometry, vol. 1. Second edition. Springer-Verlag, Berlin. Translated from the 1988 Russian edition and with notes by Miles Reid. Tabera, L.F., 2007. Two tools in algebraic geometry: construction of configurations in tropical geometry and hypercircles for the simplification of parametric curves. PhD thesis,Universidad de Cantabria, Université de Rennes I. Villarino, C. 2007. Algoritmos de optimalidad algebraica y de cuasi-polinomialidad para curvas racionales. PhD thesis, Universidad de Alcalá.