Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables

TL;DR

This paper investigates algebraic and Puiseux series solutions of autonomous algebraic ODE systems of dimension one, providing methods to decide existence and compute all algebraic solutions algorithmically.

Contribution

It introduces a notion of algebraic dimension for such systems and develops an algorithm to compute all algebraic solutions, including their approximation by convergent series.

Findings

All formal Puiseux series solutions can be approximated by convergent solutions.

Existence of Puiseux series and algebraic solutions can be decided algorithmically.

A symbolic algorithm for computing all algebraic solutions is presented.

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.