Unirational differential curves and differential rational parametrizations

In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper differential rational parametrization for the differential curve.

[1]  J. Ritt,et al.  Differential Equations from the Algebraic Standpoint , 1933, Nature.

[2]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[3]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[4]  Franz Winkler,et al.  Algebraic General Solutions of First Order Algebraic ODEs , 2015, CASC.

[5]  E. R. Kolchin.  Extensions of differential fields. III , 1947 .

[6]  E. Kolchin.  Differential Algebra and Algebraic Groups , 2012 .

[7]  Sonia L. Rueda.  A perturbed differential resultant based implicitization algorithm for linear DPPEs , 2011, J. Symb. Comput..

[8]  E. R. Kolchin,et al.  Extensions of Differential Fields, II , 1942 .

[9]  Franz Winkler,et al.  Rational general solutions of planar rational systems of autonomous ODEs☆ , 2011, J. Symb. Comput..

[10]  Y. Andre,et al.  Differential algebra , 2020, De Rham Cohomology of Differential Modules on Algebraic Varieties.

[11]  William Y. Sit.  THE RITT–KOLCHIN THEORY FOR DIFFERENTIAL POLYNOMIALS , 2002 .

[12]  Josef Schicho,et al.  Rational Parametrization of Surfaces , 1998, J. Symb. Comput..

[13]  Xiao-Shan Gao,et al.  Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form , 2010, Transactions of the American Mathematical Society.

[14]  Giuseppa Carrà Ferro.  A Resultant Theory for Ordinary Algebraic Differential Equations , 1997, AAECC.

[15]  Erich Kaltofen,et al.  Polynomial factorization: a success story , 2003, ISSAC '03.

[16]  Franz Winkler,et al.  The Algebro-Geometric Method for Solving Algebraic Differential Equations — A Survey , 2019, Journal of Systems Science and Complexity.

[17]  Evelyne Hubert,et al.  Resolvent Representation for Regular Differential Ideals , 2003, Applicable Algebra in Engineering, Communication and Computing.

[18]  Manuel Bronstein,et al.  An Introduction to Pseudo-Linear Algebra , 1996, Theor. Comput. Sci..

[19]  Franz Winkler,et al.  Rational general solutions of first order non-autonomous parametrizable ODEs , 2010, J. Symb. Comput..

[20]  Bruno Buchberger,et al.  Applications of Gröbner Bases in Non-linear Computational Geometry , 1987, Trends in Computer Algebra.

[21]  J. Rafael Sendra,et al.  Linear complete differential resultants and the implicitization of linear DPPEs , 2010, J. Symb. Comput..

[22]  Marc Chardin,et al.  Differential Resultants and Subresultants , 1991, FCT.

[23]  Ron Goldman,et al.  Implicitizing Rational Curves by the Method of Moving Algebraic Curves , 1997, J. Symb. Comput..

[24]  Bruno Buchberger,et al.  Applications of Gro¨bner bases in non-linear computational geometry , 1988 .

[25]  J. Rafael Sendra,et al.  Symbolic Parametrization of Curves , 1991, J. Symb. Comput..

[26]  R. J. Walker.  Algebraic curves , 1950 .

[27]  J. Rafael Sendra,et al.  Rational general solutions of systems of first-order algebraic partial differential equations , 2018, J. Comput. Appl. Math..

[28]  J. Rafael Sendra,et al.  A solution method for autonomous first-order algebraic partial differential equations , 2016, J. Comput. Appl. Math..

[29]  J. Rafael Sendra,et al.  Tracing index of rational curve parametrizations , 2001, Comput. Aided Geom. Des..

[30]  Xiao-Shan Gao,et al.  Implicitization of differential rational parametric equations , 2003, J. Symb. Comput..

[31]  Xiao-Shan Gao,et al.  Implicitization of Rational Parametric Equations , 1992, J. Symb. Comput..

[32]  J. Rafael Sendra,et al.  Rational Algebraic Curves: A Computer Algebra Approach , 2007 .

[33]  Gleb Pogudin,et al.  Bounds for Elimination of Unknowns in Systems of Differential-Algebraic Equations , 2016 .

[34]  Xiao-Shan Gao,et al.  A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs , 2006, J. Symb. Comput..

[35]  Franz Winkler,et al.  Rational general solutions of higher order algebraic odes , 2013, J. Syst. Sci. Complex..

[37]  Joseph Johnson,et al.  Kahler Differentials and Differential Algebra , 1969 .

[38]  Xiaoshan,et al.  LUEROTH’S THEOREM IN DIFFERENTIAL FIELDS , 2002 .

[39]  T. Glad,et al.  An Algebraic Approach to Linear and Nonlinear Control , 1993 .