On the complexity of solving ordinary differential equations in terms of Puiseux series

TL;DR

This paper establishes that solving polynomial differential equations using Puiseux series can be done with single exponential complexity relative to the number of series terms, extending previous bounds to all differential polynomials.

Contribution

It generalizes the complexity bound for solving differential equations in Puiseux series from Riccati to all polynomial differential equations using a differential Newton-Puiseux algorithm.

Findings

Binary complexity is single exponential in the number of series terms.

Extends previous bounds from Riccati equations to all polynomial differential equations.

Introduces a differential Newton-Puiseux procedure for algebraic equations.

Abstract

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.