Algorithms and Methods in Differential Algebra

Abstract Founded by J.F. Ritt, Differential Algebra is a true part of Algebra so that constructive and algorithmic problems and methods appear in this field. In this talk, I do not intend to give an exhaustive survey of algorithmic aspects of Differential Algebra but I only propose some examples to give an insight of the state of knowledge in this domain. Some problems are known to have an effective solution, others have an efficient effective solution which is implemented in recent computer algebra systems, and the decidability of some others is still an open question, which does not prevent computations from leading to interesting results. Liouville's theory of integration in finite terms and Risch's theorem are examples of problems that computer algebra systems now deal with very efficiently (implementation work by M Bronstein). In what concerns linear differential equations of arbitrary order, a basis for the vector space of all liouvillian solutions can “in principle” be computed effectively thanks to a theorem of Singer's [17, 22]; the complexity bound is actually awful and a lot of work is done or in progress, especially by M. Singer and F. Ulmer, to give realistic algorithms [20, 21] for third-order linear differential equations. Existence of liouvillian first integrals is a way to make precise the notion of integrability of vector fields. Even in the simplest case of three-dimensional polynomial vector fields, no decision procedure is known for this existence. Nevertheless, explicit computations with computer algebra yield interesting solutions for special examples. In this case, the process of looking for so-called Darboux curves can only be called a method but not an algorithm; for a given degree, this search is a classical algebraic elimination process but no bound is known on the degree of the candidate polynomials. This paper insists on the search of liouvillian first integrals of polynomial vector fields and a new result is given: the generic absence of such liouvillian first integrals for factorisable polynomial vector fields in three variables.