關於零點定理的一些討論

Abstract

在第三節中,我們給了兩個比較基本的零點定理的証明。在第四節中,我們討論了不同形式的零點定理的相互關係, 並且我們也給了一些不同形式的零點定理的不同的証明。 Alon 是組合的零點定理的主創者, 這個定理可以可以用來證明另一個定理 (定理13)並且在組合學和數輪上有很多的應用。 在本篇論文中,我們用另外一個觀點來證明此定理並將它推廣(定理12)。在例子 14和例子 15 中,我們也給了一些應用。

In Section 4 we give two elementary proofs of Nullstellensatz.The first is due to Enrique Arrondo which is more brief. The second is due to Terrance Tao which is constructive.In Section 5 we will discuss different proofs of above theorems. The first is above three forms which are equivalent. The second we give two different proofs of strong form. The third we give four differentproofs of field form. Alon is the principal founder of the Combinatorial Nullstellensatz. This theorem can prove another theorem (see Theorem 13) which has many applications in combinatorics and number theory. In this paper we will give another view point to this proof and generalize this theorem (see Theorem 12). We also give some applications in Example 14 and Example 15.