Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables. (January 2023)

• Record Type: Journal Article
• Authors: Cano, José; Falkensteiner, Sebastian; Robertz, Daniel; Sendra, J.Rafael
• Is Part Of: Journal of symbolic computation. Volume 114(2023)
• Page Start: 1
• Page End: 17
• Publication Date: 2023-01
• Subjects: Algebraic autonomous ordinary differential equation -- Puiseux series solution -- Convergent solution -- Artin approximation -- Algebraic solution -- Thomas decomposition; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.04.012
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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.