APPROXIMATION THEORY IN LOCAL RINGS
Abstract
Approximation in local rings was developed by many mathematicians. M. J. Greenberg proved an approximation theorem for valuation rings by algebraic geometric methods; M. Artin proved two important approximation theorems in higher dimension by algebraic geometric methods. J. Becker, J. Denef, L. Lipshitz and L. van den Dries proved the above results and extensions by model theoretic methods. Of course, many other results about approximation in local ring were proved over these years. In this thesis, we prove some generalized results about approximation theorems in local rings, and give some bounds for the approximation functions. The main tools for the proofs in this thesis are the following: (i) The model theoretic method made of J. Becker, J. Denef, L. Lipshitz and L. van den Dries; (ii) The induction on the number of unknowns and the rank of the system of polynomial equations introduced by M. Kneser and also by J. Denef and L. Lipshitz; (iii) The combination of implicit function theorem and Weierstrass Preparation Theorem used by M. Artin; (iv) The application of Galois theory to approximation theory made by B. J. Birch and K. M('c)Cann.
Degree
Ph.D.
Subject Area
Mathematics
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