Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

• Published in: Mathematics in computer science Vol. 15; no. 2; pp. 189 - 198
• Main Authors: Cano, José, Falkensteiner, Sebastian, Sendra, J. Rafael
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2021; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Convergence; Differential equations; Mathematics; Mathematics and Statistics; Ordinary differential equations; Formal Puiseux series solution; 34A05; Rational solutions; Algebraic autonomous ordinary differential equation; 34A34; Convergent solution; 34M25; 14H50; Algebraic space curve; Algebraic solutions
• ISSN: 1661-8270; 1661-8289
• DOI: 10.1007/s11786-020-00478-w

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
)
∈
C
2
, we prove the existence of a convergent Puiseux series solution
y
(
x
) of the original system such that
y
(
x
0
)
=
y
0
.