Holomorphic functions with bounds

Let P be a pq matrix of polynomials in n complex variables. If Ω is a domain of holomorphy in complex n-space and if f is a p-tuple of holomorphic functions in Ω and if the system of linear equations Pu=f has a solution u which is a holomorphic q-tuple in Ω, then it has a solution which satisfies an estimate in terms of f. The paper discusses in outline the proof of such a result published by the author. The discussion is motivated by a simple illustration in the case p=q=1. In addition the proof is given of a theorem on trivial cohomology with bounds announced without proof in the author's paper mentioned above. As an application of his main result, the author generalizes to systems a theorem of Komec concerning the possibility of solving constant coefficient partial differential equations where the solution is required to be a temperate distribution with support in a cone. (author)