ON HIGHER ORDER DIFFERENTIAL IDEALS

Let M be a smooth m-dimensional manifold, and (C-k,C-n M)(0), n <= m, the manifold of co-contact of order k and codimension n over M. Given a differential system (W-k)(0) subset of (C-k,C-n M)(0) in a neighborhood of X-0(k) is an element of(W-k)(0), we associate with (W-k)(0) an ideal Y-k (W-k)(0) definite in a open of (C-k-1,C-n M)(0). We prove that Y-k (W-k)(0) is an ideal locally generated by a set of independent 1-forms. It coincides locally with the ideal in Lambda* (C-k-1,C-n M)(0) generated by the set tau(1)(k-1) = {omega is an element of Lambda(1) (C-k-1,C-n M)(0)/i(X)omega = 0, X is an element of tau chi(k-1) (M)}. Here, tau chi(k-1) (M) is the submodule of vector-fields in (C-k-1,C-n M)(0) lying in the distribution that every point theta(k-1) of an open in (C-k-1,C-n M)(0) associates with the n-dimensional R-plane L-theta k, at theta(k-1). Moreover, conditions on the differential system (W-k)0 are given for the ideal Y-k (W-k)0 to be a differential closed.