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The Jacobian Problem concerning Polynomial mappings

关于多项式映射的雅可比问题

基本信息

批准号：
06804003
负责人：
KAMBAYASHI Tatsuji
金额：
$ 1.22万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

项目摘要

The long-outstanding Jacobian Problem (abbrev. (JP)) asks the following question : If a polynomial endomorphism of the complex affine n-space is locally invertible every-where, is it then true that such an endomorphism is globally invertible and, therefore, an automorphism of the whole space? While the answer is widely believed to be in the affirmative, no one so far has been able to prove this is so, even when dimension n=2.Supported by the present three-year grant, our effort since 1994 toward solving (JP) in the positive direction has been made in the framework of infinite-dimensional algebras and varieties, called pro-affine algebras and ind-affine varieties in our work. The monoid U of all principal endormorphisms with Jabobian determinant=1, and the group G of all principal automorphisms are both ind-affine varieties, and there is a natural embedding G*U.The first half of the grant period was spent for (a) founding the theory of pro-affine algebras and ind-affine vareities, (b) proving that, if one can say the embedding G*U is a closed map, then G=U (i.e., an affirmative resolution of (JP) is obtained, and (c) finding a number of conditions each of which sufficient for the closedness of the map in question. The results (a), (b), (c) have now been published in our paper : "Pro-affine alg ebras, ind-affine groups and the Jacobian Problem, "Journal of Algebra, vol.185 (1996), 481-501.In the latter half of the period we have attempted to prove any one of the conditions mentioned in (c) above, but have actually ended up getting a counter-example to one of those and negative prospects for the others. On a more positve side, though, we have found that proving the embedding G*U to be a locally open map suffices for the desired solution of (JP). This direction has been found to require deepening of our pro-affine/ind-affine theory, such as the review of our topology and a new definition of etale maps in our category. Our research is strongly progressing in this.

长期未解决的雅可比问题（焦v.（JP））提出了以下问题：如果复仿射n-空间的多项式自同态处处局部可逆，那么这样的自同态全局可逆，因此是整个空间的自同构是真的吗？虽然人们普遍认为答案是肯定的，但迄今为止还没有人能够证明这一点，甚至当维数n= 2时也是如此。在目前三年的资助下，我们自1994年以来一直在无限维代数和簇的框架下（在我们的工作中称为pro-affine代数和ind-affine簇）朝着正方向求解（JP）。所有Jabobian行列式=1的主自同态的幺半群U和所有主自同构的群G都是ind-affine簇，并且存在自然嵌入G*U。授予期间的前半部分用于（a）建立亲仿射代数和ind-affine簇的理论，（B）证明，如果可以说嵌入G*U是闭映射，则G=U（即，得到（JP）的一个肯定解，以及（c）找到若干条件，每个条件都足以证明所讨论的映射的闭性。结果（a）、（B）、（c）现已发表在我们的论文中：“Pro-affine algebras，ind-affine groups and the Jacobian Problem，“Journal of Algebra，vol.185（1996），481- 501.在后半期，我们试图证明上面（c）中提到的任何一个条件，但实际上最终得到了其中之一的反例和其他人的负面前景。然而，在更积极的一面，我们发现证明嵌入G*U是局部开映射就足以得到（JP）的理想解。这个方向已经发现，需要深化我们的亲仿射/ind-affine理论，如审查我们的拓扑结构和一个新的定义etale地图在我们的类别。我们的研究在这方面取得了很大进展。