Theory of resolution of singularities in positive characteristic

这一研究课题多年来一直是世界数学界关注的课题之一。根据我们的常识，这一定是一门很难的学科。我知道学习这门学科要比学习标准学科花的时间长得多。这是因为我们需要非常精确的参数来操纵复杂的情况，避免隐藏的陷阱。在项目期间，我从不同的角度考虑了这个主题的各个方面。几年前，我成功地构建了三变量理论。(然而，这似乎与几十年前广中所做的研究结果基本相同。Hironaka的结果没有正式发表。这是某本书的附录。在这里，我想解释由四变量幂级数定义的超曲面奇点情况的最新结果，这是最简单的情况，没有明确的正结果。它解释了这门课的全部内容。利用三个变量的情况下的想法，我构建了一个更复杂的四变量情况的理论。在正特征情况下，对维数或变量数的归纳法不起作用。毫无疑问，对变量数量的归纳法是Hironaka在特征零方面取得成功的关键。因此，四个变量的情况与三个变量的情况有本质的区别。在正特征情况下，我们定义了与变量数顺序相同的牛顿多边形数，并且我们对牛顿多边形的顺序使用归纳法而不是对变量数进行归纳法。我们也可以将这种方法应用于特征零点，并且我们可以得出在特征零点奇异点的解析总是可能的结论。通过这种方法，我们知道，在几乎所有的情况下，在正特征中几乎都发生与特征为零时几乎相同的现象。但在特征性阳性中出现非常病理现象的病例很少。使这些病理病例变得更加珍贵是我们的基本课题。我惊讶地发现，除了几年前我发现的三个病例外，可能还存在几个病理病例。我又发现了三个案例。值得庆幸的是，我们只分析了6个案例，就得出了最终的结果，取得了很大的进步。少

This research subject is one of subjects of the world-wide mathematical society for long years. According to our common sense, it must be a very difficult subject. I know that it takes by far longer time to study this subject than standard ones. It is because we need very exact arguments to manipulate complicated situation avoiding hidden traps.During the term of the project I considered various aspects of the subject from various view points. Several years ago I succeeded to construct the theory of three variables. (However, it seems to be essentially the same result by Hironaka several tens years ago. Hironaka's result was not published formally. It is an appendix of some book.) Here, I would like to explain the latest result for the case of hypersurface singularity defined by a power series with four variables, which is the simplest case with no definite positive results. It explains whole scheme of the subject. Using ideas in the case of three variables, I constructed the theory of … More the more complicated case of four variables. In positive characteristic cases induction on the dimension or on the number of variables does not work. Needless to say, induction on the number of variables is the key of Hironaka's success in characteristic zero. Therefore, there is essential difference between the case of four variables and the case of three variables. In positive characteristic case we define the same number of Newton polygons in order as the number of variables, and we use induction on the order of Newton polygons instead of induction on the number of variables. We can apply this method also in characteristic zero, and we can conclude that resolution of singularities is always possible in characteristic zero. By this method one knows that also in positive characteristic almost the same phenomena as in characteristic zero occur in almost all cases. However, there exist few cases where very pathological phenomena occur in characteristic positive. It is our essential subject to make these pathological cases dearer. I surprised to know that there may exist several pathological cases in addition to three cases I found some years ago. I found three cases further. It is dear that analyzing only six cases, we can reach the final result, and great advance has been achieved. Less