Essential Components of an Algebraic Differential Equation

We present an algorithm to determine the set of essential singular solutions of a differential equation. Essential singular solutions can be informally described as follows: the general solution of a differential equation is usually described as a solution depending on a number of arbitrary constants equal to the order of the differential equation. The essential singular solutions are those that cannot be obtained by substituting numerical values to the arbitrary constants in the general solution.† Adherence, defined in Ritt (1950, VI.2), is the correct concept: singular solutions that are not essential are adherent to the general solution or to one of the essential singular solutions. For first-order differential equations, Hamburger (1893) showed that the essential singular solutions were envelopes of the family of curves given by the general solution. Ritt gave a similar result for first-order partial differential equations (Ritt, 1945a) and for special cases of second-order differential equations (Ritt, 1946). These analytic and geometric properties may be seen as a first application for our algorithm. Nonetheless, the concepts involved translate into algebraic definitions and properties. We shall thus work in the frame of differential algebra. Central there is the definition of the general solution due to Ritt (1930). A system of algebraic differential equations can be seen as a set Σ of differential polynomials in an appropriate differential polynomial ring. The radical differential ideal generated by Σ can be written as the irredundant intersection of a finite number of prime differential ideals called the components of the radical differential ideal. In the case Σ consists of a single differential polynomial that is irreducible when considered as a polynomial, one of these components defines the general solution. The others are essential singular components. For our purpose, we will extend the definition of the general component to regular differential polynomials. Regular differential polynomials arise in a practical algorithm