In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential elimination and analyze the behavior of the dimension in the resulting Thomas decomposition. For such systems of algebraic dimension one, we show that all formal Puiseux series solutions can be approximated up to an arbitrary order by convergent solutions. We show that the existence of Puiseux series and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. The output can either be represented by triangular systems or by their minimal polynomials.

翻译：在本文中,我们用几种不同的定点来研究自主代数极值的系统。 我们通过将这种系统视为代数系统来开发代数维度的概念。 然后,我们应用差别消除法并分析由此产生的托马斯分解过程的维度行为。 对于这种代数维度系统,我们显示,所有正式的Puiseux系列解决方案都可以通过趋同式解决方案来近似于任意顺序。 我们显示,Puiseux系列和代数解决方案的存在可以通过算法来决定。 此外,我们提出了一个象征性的算法来计算所有代数解决方案。 输出可以由三角系统或以其最小的多元模型来代表。