Gröbner fan and universal characteristic sets of prime differential ideals

Abstract

The concepts of Gröbner cone, Gröbner fan, and universal Gröbner basis are generalized to the case of characteristic sets of prime differential ideals. It is shown that for each cone there exists a set of polynomials which is characteristic for every ranking from this cone; this set is called a strong characteristic set, and an algorithm for its construction is given. Next, it is shown that the set of all differential Gröbner cones is finite for any differential ideal. A subset of the ideal is called its universal characteristic set, if it contains a characteristic set of the ideal w.r.t. any ranking. It is shown that every prime differential ideal has a finite universal characteristic set, and an algorithm for its construction is given. The question of minimality of this set is addressed in an example. The example also suggests that construction of a universal characteristic set can help in solving a system of nonlinear PDE’s, as well as maybe providing a means for more efficient parallel computation of characteristic sets.