Matrix polynomials and the index problem for elliptic systems

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.'s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.'s.