仿射環與冪級數環上的正規化定理

Abstract

在第一節，我們給了一些正規化定理的應用和它的幾何意義。另外，我們敘述了這篇論文各章節所有主要的結果。從第二節到第五節，我們討論基本的正規化定理和一些特殊版本。在第六節，我們給了冪級數環上的正規化定理。在第七節中，我們先給了三種 Weierstrass Proposition Theorem 的證明。其中一個為我們可操作的長除法證明；而另兩個可以得到更強的結果。最後，我們使用 Weierstrass Proposition Theorem 去重新證明第六節的正規化定理，並更進一步的，我們可以證明在收斂冪級數環上的正規化定理。

In Section 1, we give some applications of the normalization theorem and the geometric significance of the normalization theorem. Moreover, we state all main results among this paper. From Section 2 to Section 5, we discuss the basic normalization theorem and more special versions. In Section 6, we give the classical normalization theorem for power series rings. In Section 7, we give three proofs for Weierstrass Preparation Theorem. One of them is the long division algorithm and the others give us the stronger result. At last, we use the Weierstrass Preparation Theorem to reprove the normalization theorem for power series rings and for convergent power series rings.