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Descent of Ordinary Differential Equations with Rational General Solutions

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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.

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shuang Feng & Ruyong Feng

  2. School of Mathematics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Shuang Feng & Ruyong Feng

Authors
  1. Shuang Feng
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  2. Ruyong Feng
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Corresponding authors

Correspondence to Shuang Feng or Ruyong Feng.

Additional information

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.

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Feng, S., Feng, R. Descent of Ordinary Differential Equations with Rational General Solutions. J Syst Sci Complex 33, 2114–2123 (2020). https://doi.org/10.1007/s11424-020-9310-x

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  • DOI: https://doi.org/10.1007/s11424-020-9310-x

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