# Symbolic Computations of First Integrals for Polynomial Vector Fields

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 authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.
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Preprints, Working Papers, ...
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Cited literature [5 references]

https://hal.archives-ouvertes.fr/hal-01619911
Contributor : Guillaume Chèze <>
Submitted on : Tuesday, December 18, 2018 - 11:31:33 AM
Last modification on : Monday, April 29, 2019 - 4:59:17 PM
Long-term archiving on : Wednesday, March 20, 2019 - 11:49:09 AM

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### Identifiers

• HAL Id : hal-01619911, version 2
• ARXIV : 1710.08225

### Citation

Guillaume Chèze, Thierry Combot. Symbolic Computations of First Integrals for Polynomial Vector Fields. 2018. ⟨hal-01619911v2⟩

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