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Symbolic Computations of First Integrals for Polynomial Vector Fields

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Abstract

In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in \(\tilde{\mathcal {O}}(N^{\omega +1})\), where N is the bound on the degree of a representation of the first integral and \(\omega \in [2;3]\) is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors’ websites. In the last section, we give some examples showing the efficiency of these algorithms.

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Acknowledgements

Guillaume Chèze acknowledges the support and hospitality of the Laboratorio Fibonacci (Pisa) where a part of this work has been developed during November 2016 when he was visiting Thierry Combot. The authors thank the anonymous reviewers for their careful reading of our manuscript and for pointing out the reference [15].

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Correspondence to Guillaume Chèze.

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Communicated by Evelyne Hubert.

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Chèze, G., Combot, T. Symbolic Computations of First Integrals for Polynomial Vector Fields. Found Comput Math 20, 681–752 (2020). https://doi.org/10.1007/s10208-019-09437-9

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