The Algebro-Geometric Method for Solving Algebraic Differential Equations——A Survey

This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations(ADEs). An algebraic differential equation is a polynomial relation between a function, some of its partial derivatives, and the variables in which the function is de?ned.Regarding all these quantities as unrelated variables, the polynomial relation leads to an algebraic relation de?ning a hypersurface on which the solution is to be found. A solution in a certain class of functions, such as rational or algebraic functions, determines a parametrization of the hypersurface in this class. So in the algebro-geometric method the author ?rst decides whether a given ADE can be parametrized with functions from a given class; and in the second step the author tries to transform a parametrization into one respecting also the differential conditions. This approach is relatively well understood for rational and algebraic solutions of single algebraic ordinary differential equations(AODEs). First steps are taken in a generalization to systems and to partial differential equations.