Using Algebraic Geometry for Solving Differential Equations

Given a first order autonomous algebraic ordinary differential equation, i.e. an equation of the form $F(y,y')=0$ with polynomial $F$ and complex coefficients, we present a method to compute all formal Puiseux series solutions. By considering $y$ and $y'$ as independent variables, results from Algebraic Geometry can be applied to the implicitly defined plane curve. This leads to a complete characterization of initial values with respect to the number of distinct solutions extending them. Furthermore, the computed formal solutions are convergent in suitable neighborhoods and for any given point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point. Recently we generalized some of these results to systems of ODEs which implicitly define a space curve and to differential equations of the form $F(y,y^{(r)})=0$. Moreover, we present a method to compute local solutions with real or even rational coefficients only.

This is a joint work with Jose Cano, Rafael Sendra and Daniel Robertz.