[Paper Review] Existence and convergence of Puiseux series solutions for first order autonomous differential equations

José Cano, Sebastian Falkensteiner+1 more|arXiv (Cornell University)|Aug 24, 2019

Advanced Differential Equations and Dynamical Systems7 references6 citations

TL;DR

This paper proves that all formal Puiseux series solutions to first-order autonomous ODEs F(y, y') = 0 are convergent, both around finite points and at infinity. The authors present a constructive algorithm to compute these solutions and demonstrate that every point in the complex plane lies on an analytic solution curve, establishing global existence and convergence of Puiseux-type solutions.

ABSTRACT

Given an algebraic first order autonomous ordinary differential equation F(y,y')=0, we prove that every formal Puiseux series solution of F(y,y')=0, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

Motivation & Objective

• To establish the convergence of all formal Puiseux series solutions to first-order autonomous ODEs F(y, y') = 0 at any finite point or at infinity.
• To develop a constructive algorithm for computing all such Puiseux series solutions explicitly.
• To prove that for any point in the complex plane, there exists a solution curve passing through it that is analytic and defined by a Puiseux series.
• To bridge the gap between formal power series solutions and convergent analytic solutions in the context of autonomous algebraic ODEs.

Proposed method

• Utilizes algebraic geometry techniques to analyze the structure of the algebraic curve F(y, y') = 0 in the (y, y')-plane.
• Applies the theory of Puiseux series expansions to parametrize branches of the algebraic curve near singular or regular points.
• Employs a recursive algorithmic procedure to compute the coefficients of the Puiseux series, ensuring convergence through careful control of the growth rate of coefficients.
• Leverages the autonomous nature of the ODE (no explicit dependence on the independent variable) to reduce the problem to a curve-fitting and parametrization task.
• Applies convergence criteria from complex analysis to show that the constructed Puiseux series solutions converge in a punctured neighborhood of the expansion point.
• Demonstrates that the solution curves are analytic and can be extended to cover any point in the complex plane via appropriate parametrization.

Experimental results

Research questions

• RQ1 Are all formal Puiseux series solutions to first-order autonomous ODEs F(y, y') = 0 convergent?
• RQ2 Can a systematic algorithm be constructed to compute all such Puiseux series solutions?
• RQ3 Does every point in the complex plane lie on an analytic solution curve defined by a Puiseux series?
• RQ4 What structural properties of the algebraic curve F(y, y') = 0 ensure the convergence of its Puiseux parametrizations?
• RQ5 How does the autonomous nature of the ODE influence the existence and convergence of Puiseux solutions?

Key findings

• All formal Puiseux series solutions to F(y, y') = 0 are convergent, both at finite points and at infinity.
• A constructive algorithm is provided to compute all such Puiseux series solutions explicitly.
• For any point in the complex plane, there exists a solution curve passing through it that is analytic and represented by a Puiseux series.
• The convergence of Puiseux series solutions is guaranteed by the algebraic and autonomous structure of the ODE.
• The solution curves form a global foliation of the complex plane by analytic curves, each defined by a convergent Puiseux series.
• The method ensures that the Puiseux series solutions are not only formal but also represent actual convergent solutions in a neighborhood of the expansion point.