Puiseux Series Solutions of Ordinary Polynomial Differential Equations : Complexity Study
نویسنده
چکیده
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 in 1990 by Grigoriev 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. 2000 Mathematics Subject Classification: 12H05, 13F25, 68W30, 68Q25.
منابع مشابه
On the complexity of solving ordinary differential equations in terms of Puiseux series
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 algori...
متن کاملA note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation
We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the complexity of solving arbitrary ordinary polynomial differential equations in terms of Puiseux series. Introduction LetK = Q(T1, . . . , Tl)[η] be a finite extens...
متن کاملConstruction of Special Solutions for Nonintegrable Systems
The Painlevé test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a search for so...
متن کاملFormal Power Series Solutions of Algebraic Ordinary Differential Equations
In this paper, we consider nonlinear algebraic ordinary differential equations (AODEs) and study their formal power series solutions. Our method is inherited from Lemma 2.2 in [J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Mathematische Annalen, 267(1984), 213-238] for expressing high order derivatives of a differential polynomial via their lower order on...
متن کاملLaurent Series Solutions of Algebraic Ordinary Differential Equations
This paper concerns Laurent series solutions of algebraic ordinary differential equations (AODEs). We first present several approaches to compute formal power series solutions of a given AODE. Then we determine a bound for orders of its Laurent series solutions. Using the order bound, one can transform a given AODE into a new one whose Laurent series solutions are only formal power series. The ...
متن کامل