论文标题
自主代数odes odes of dimension的代数，理性和PUISE串联系列解决方案
Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one
论文作者
论文摘要
在本文中，我们研究了某些类型的自主差微分方程系统的代数，理性和正式PUISEUX系列解决方案。更确切地说，我们处理与代数集相关的系统是维数。我们在系统的解决方案与相关的一阶自主差异方程式的解决方案之间建立了一种关系，我们称之为减少的微分方程。使用此类方程式的结果，我们证明了系统的正式PUISEUX系列解决方案的收敛性，在有限点或无穷大范围内扩展，并提出了一种描述它们的算法。此外，我们绑定了可能的代数和理性解决方案的程度，并提供了一种算法来决定其存在并在此类解决方案存在的情况下计算它们。此外，如果还要降低的微分方程不是微不足道的，那么对于复杂平面中的每个给定点$（x_0，y_0）$，我们证明了原始系统的收敛puiseux系列解决方案$ y（x）$，因此$ y（x_0）= y__0 $。
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point $(x_0,y_0)$ in the complex plane, we prove the existence of a convergent Puiseux series solution $y(x)$ of the original system such that $y(x_0)=y_0$.
