Restrictions on the coefficients of hyperbolic systems of partial differential equations

The paper deals with hyperbolic homogeneous systems [Formula: see text] of partial differential equations with constant coefficients for an N-vector u(t,x₁,...,x_n). Here, P is a matrix form of order N and degree m. In the scalar case (N = 1), every hyperbolic P is limit of strictly hyperbolic ones. This does not hold for systems as is shown here for the special case N = n = 3, m = 2. Assuming P(1,0,...,0) to be the unit matrix, we represent P by a point in R₈₁. The hyperbolic P form a closed set H in R₈₁, the strictly hyperbolic ones an open subset H_s of H. An example is given for a P in H which is not in the closure of H_s. Moreover, it is shown that near that P the set H coincides with an algebraic manifold of codimension 4.