Theory of resolution of singularities in positive characteristic

这一研究课题是世界数学界长期关注的课题之一。根据我们的常识，这一定是一个非常困难的问题。我知道学习这门学科比学习标准学科需要更长的时间。这是因为我们需要非常精确的论据来操纵复杂的情况，避免隐藏的陷阱。在项目期间，我从不同的角度考虑了这个问题的各个方面。几年前，我成功地建立了三变量理论。（然而，这似乎与几十年前Hironaka的结果基本相同。Hironaka的结果没有正式发表。这是一本书的附录）。在这里，我想解释一下由四个变量的幂级数定义的超曲面奇异性的最新结果，这是最简单的情况，没有明确的正结果。它解释了这个主题的整个计划。利用三个变量的情况下的想法，我构建了理论， ...更多信息 更复杂的四个变量。在正特征的情况下，对维度或变量数的归纳不起作用。不用说，对变量个数的归纳是Hironaka在特征零点上成功的关键。因此，四个变量的情况与三个变量的情况之间存在本质区别。在正特征情形下，我们定义了与变量个数相同的牛顿多边形的阶数，并且我们用牛顿多边形阶的归纳法代替了变量个数的归纳法。我们也可以将这种方法应用于特征零点，我们可以得出结论，奇异性的解决方案总是可能的特征零点。通过这种方法，人们知道，在正特性中，几乎在所有情况下都会出现与零特性中几乎相同的现象。然而，很少有病例在特征性阳性中发生非常病理性的现象。使这些病态的病例变得更加珍贵是我们的基本课题。我很惊讶地知道，除了我几年前发现的三个病例外，还可能存在几个病理病例。我又发现了三个案子。通过对六个案例的分析，得出了最后的结论，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