Factorization-free Decomposition Algorithms in Differential Algebra

This paper makes a contribution to differential elimination and more precisely to the problem of computing a representation of the radical differential ideal generated by a system of differential equations (ordinary or partial). The approach we use here involves characteristic set techniques;‡ these were introduced by J. F. Ritt, the founder of differential algebra. The basic idea is to write the radical differential ideal generated by a finite set Σ of differential polynomials as an intersection of differential ideals that are uniquely defined by their characteristic sets. We will call these latter differential ideals components of {Σ} and we will call their intersection the characteristic decomposition of {Σ}. With a characteristic decomposition of {Σ} we can determine whether Σ = 0 has any solution, test membership to {Σ} and study the dimension properties of {Σ}. Similar to algebraic elimination methods, by choosing an appropriate ranking, an algorithm to compute a characteristic decomposition can also answer questions like: do the solutions of Σ = 0 satisfy: — An algebraic equation? Find all such constraints. — An ordinary differential equation in one of the independent variables? Find these ordinary differential equations. — A differential equation involving only a specific subset of the dependent variables? Find these differential equations. Algorithms in differential elimination have been applied in symmetry analysis of partial differential equations (Clarkson and Mansfield, 1994; Mansfield et al., 1998) and control theory (Diop, 1991, 1992; Fliess and Glad, 1993). Ritt (1950) gave an algorithm to compute a characteristic decomposition of {Σ} where