Abstract
We define and discuss equations on Banach rings (algebras) which are of polynomial form. We prove a local uniqueness theorem for the homogeneous case, and an existence and local uniqueness theorem for the nonhomogeneous case. In order to apply these results to the equations of Lagrangian quantum field theory we find it necessary to extend the concept of a ring to that of an n-ring. The resulting theory is applied to a simple model equation arising in quantum field theory.
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