| Chapter 1 provides definitions and basic properties of the algebra O (S) of functions holomorphic on a subset S of Cn. I demonstrate some basic topological properties of O (S), prove a generalized form of Bers' Theorem, and thereby show that O (S) is isomorphic to O (S), where S is the holomorphic convexification of S. I also generalize some of the results of First-Order Conformal Invariants, by Joseph Becker, C Ward Henson, and Lee A. Rubel [3], to the multi-variable setting. In particular, I characterize the ideals for which the variety consist.s of a. single point, and shown that, given a generalization of the Weierstrass value theorem and a vector of functions with infinitely many discrete zeros on S, any formula of second-order arithmetic can be expressed in the first-order language of O (S).;Chapter 2 is a study of quasigraded rings and modules, and of local, effectively Noetherian rings and modules; quasigraded rings generalize the concept of graded rings to larger structures, while effectively Noetherian rings are a first-order definable counterpart of Noetherian rings. An effective version of the Hilbert Basis Theorem is proven.;Chapter 3 is a study of holomorph algebras, whose first-order properties resemble those of local power series rings. Basic results of the theory of holomorph-algebras are developed.;Chapter 4 concerns the extraction of coefficients from power series; it includes proof of the Hilbert Basis Theorem.;Chapter 5 develops a fully-local theory of varieties and attempts to use them to prove the Nullstellensatz.;Appendices on Peano Arithmetic and the theory of ultrafilters (used in constructing nonstandard models) are also included. |
