Necessary and sufficient conditions for hyperbolicity and weak hyperbolicity of systems with constant multiplicity, part I

We consider a linear system of partial differential equations, whose principal symbol is hyperbolic with characteristics of constant multiplicities. We define necessary and sufficient invariant condition in order the Cauchy problem to be well-posed in C-infinity. These conditions generalize the Levi conditions for scalar operators. The proof is based on the construction of a new non commutative determinant adapted to this case (and to the holomorphic case).