A Constructive Generalised Goursat Normal Form

We provide necessary and sufficient conditions on the derived type of a vector field distribution V in order that it be locally equivalent to a partial prolongation of the contact distribution C (1) q , on the 1 order jet bundle of maps from R to R, q ≥ 1. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is proven that if V is locally equivalent to a partial prolongation of C (1) q then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on the ambient manifold. 2000 MSC: 58J60 keywords: Goursat normal form, derived type, contact distribution, partial prolongation, local equivalence