Descent of Ordinary Differential Equations with Rational General Solutions
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Descent of Ordinary Differential Equations with Rational General Solutions∗ FENG Shuang · FENG Ruyong
DOI: 10.1007/s11424-020-9310-x Received: 8 November 2019 / Revised: 17 December 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract Let F be an irreducible differential polynomial over k(t) with k being an algebraically closed field of characteristic zero. The authors prove that F = 0 has rational general solutions if and only if the differential algebraic function field over k(t) associated to F is generated over k(t) by constants, i.e., the variety defined by F descends to a variety over k. As a consequence, the authors prove that if F is of first order and has movable singularities then F has only finitely many rational solutions. Keywords
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Algebraic ordinary differential equation, differential descent, rational general solution.
Introduction
Let K be a differential field of characteristic zero. Differential descent theory asks whether a differential algebraic variety over K can descend to an algebraic variety over CK , the field of constants of K, and it is viewed as a differential analogue of Shimura-Matsusaka theory of fields of moduli. This theory is initiated by Matsuda in [1, 2] for first order algebraic differential equations and further developed by Buium for higher order and partial differential equations. Let F be the differential algebraic function field over K associated to some irreducible differential algebraic variety. Then the descent problem is equivalent to the one asks whether F is generated over K by constants. In the case when K is an ordinary differential field and tr.deg(F /K) = 1, Matsuda in [1, 2] and Nishioka in [3] proved that F has no movable singularity if and only if there is an algebraic extension L of K such that F (L) is generated over L by constants. In [4], Buium proved the higher dimension and partial differential version of the results of Matsuda and Nishioka. FENG Shuang · FENG Ruyong KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; School of Mathematics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]; [email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grants Nos. 11771433 and 11688101, and Beijing Natural Science Foundation under Grants No. Z190004. This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
FENG SHUANG · FENG RUYONG
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This paper is mainly concerned about algebraic ordinary differential equations with rational general solutions. Rational solutions are of special interest in the community of symbolic computation. Algorithms have already been well-developed for linear differential equations (e.g., see [5–8] ). However, the situation is quite different in the case of nonlinear differential equations. Although a few algorithms have been developed to deal with the equations of special types, there is no complete algorithm to find all rational sol
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