Differential Invariants

. This paper summarizes recent results on the number and characterization of differential invariants of transformation groups. Generalizations of theorems due to Ovsiannikov and to M. Green are presented, as well as a new approach to finding bounds on the number of independent differential invariants. Consider a group of transformations acting on a jet space coordinatized by the independent variables, the dependent variables, and their derivatives. Scalar functions which are not affected by the group transformations are known as differential invariants. Their importance was emphasized by Sophus Lie, [9], who showed that every invariant system of differential equations, [10], and every invariant variational problem, [11], could be directly expressed in terms of the differential invariants. As such they form the basic building blocks of many physical theories, where one begins by postulating the invariance of the equations or the variational principle under a prescribed symmetry group. L..