Number Theory III
Number Theory III
复制标题
DOI：
10.1007/978-3-642-58227-1
复制
发表时间：
1991
期刊：
影响因子：
--
通讯作者：
S. Lang
中科院分区：
文献类型：
--
作者：
S. Lang
1988年，沙法雷维奇让我为《数学科学百科全书》写一卷关于丢番图几何的文章。我答应了，这就是音量。根据定义，丢番图问题涉及整数或有理数或各种推广形式的方程的解，如Z上的有限生成环或Q上的有限生成域。添加几何一词是为了建议几何方法。这意味着目前的卷不是基本的。有关一些基本问题的更基本方法的调查，请参见[La 9Oc]。丢番图几何领域现在发展得相当迅速。几十年前的各种猜测正在得到证实。我试图通过强调结构猜测和结果来赋予这本书某种连贯性和持久性，这样人们就可以对这一领域有一个清晰的了解。总体而言，根据百科全书的边界条件，我省略了证明。在某些情况下，当这些证明特别重要时，我确实给出了一些想法。无论如何，冗长的参考书目指的是可以找到校样的论文和书籍。我也遵循沙法雷维奇的建议来举例子，我特别选择了这些例子，这些例子展示了一些经典问题是如何被当代视觉解决的或不被解决的。费马大定理居于中间地位。虽然没有得到证明，但它不再是一个孤立的问题。
In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in sights. Fermat's last theorem occupies an intermediate position. Al though it is not proved, it is not an isolated problem any more.