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arXiv · 2412.16550

On elementary integrability of rational vector fields

Colin Christopher·Sebastian Walcher
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Abstract

We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field $K$. In view of the Prelle-Singer theorem, these are the rational vector fields that admit an elementary first integral. Elementary integrable vector fields which are not Darboux integrable -- thus the extension field is necessarily a proper extension of $K$ -- may be called exceptional by an observation in an earlier paper by Christopher et al. For dimension two we characterize all possible algebraic extension fields underlying the exceptional cases, provide a construction of all exceptional vector fields, and obtain some criteria that restrict the degree of $L$.

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Colin Christopher, Sebastian Walcher. 2024-12-21. On elementary integrability of rational vector fields. https://arxiv.org/abs/2412.16550

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