Descent of Ordinary Differential Equations with Rational General Solutions

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 )...

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Bibliographic Details
Published inJournal of systems science and complexity Vol. 33; no. 6; pp. 2114 - 2123
Main Authors Feng, Shuang, Feng, Ruyong
Format Journal Article
LanguageEnglish
Published Beijing Academy of Mathematics and Systems Science, Chinese Academy of Sciences 01.12.2020
Subjects
Complex Systems
Control
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Statistics
Systems Theory
differential descent
Algebraic ordinary differential equation
rational general solution
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Summary: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.
ISSN:1009-6124
1559-7067
DOI:10.1007/s11424-020-9310-x