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.

中文翻译：

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多项式矢量场的第一积分的符号计算

在本文中，我们展示了如何将J. Pereira引入的提取曲线推广到Darbouxian，Liouvillian和Riccati情况。通过这种方法，我们可以获得新的算法，用于计算有理，多项式平面向量场的有界度的有理，达伯先，利乌维安或里卡蒂第一个积分。我们给出概率和确定性算法。我们的概率算法的算术复杂度在\（\ tilde {\ mathcal {O}}（N ^ {\ omega +1}）\）中，其中N是第一个整数和\的表示度的界。 （\ omega \ in [2; 3] \）是线性代数的指数。该结果改进了先前的算法。我们的算法已经在Maple中实现，可以在作者的网站上找到。在最后一部分中，我们给出一些示例来说明这些算法的效率。