The Jacobian Problem concerning Polynomial mappings

长期突出的雅可比问题(简称。（JP）)提出了以下问题：如果复仿射n空间的多项式自同态在任何地方都是局部可逆的，那么这个自同态是否全局可逆，从而是整个空间的自同态？虽然人们普遍认为答案是肯定的，但到目前为止还没有人能够证明这一点，即使维度n=2也是如此。自1994年以来，我们一直在无限维代数及其变种的框架内，在目前的三年资助的支持下，在正向方向上进行求解（JP）的努力，在我们的工作中称为前仿射代数和后仿射变体。所有Jabobian行列式=1的主内同态的单群U和所有主自同态的群G都是仿射变异，并且存在一个自然嵌入G*U。资助期的前半部分用于(a)建立亲仿射代数和后仿射变量的理论，(b)证明，如果一个人可以说嵌入G*U是一个封闭映射，那么G=U（即，得到（JP）的肯定解），以及(c)找到一些条件，每个条件都足以证明所讨论的映射的封闭性。结果(a), (b), (c)已经发表在我们的论文：“前仿射代数，后仿射群和雅可比问题”，代数学报，vol.185(1996), 481-501。在这一时期的后半部分，我们试图证明上述(c)项中提到的任何一种情况，但实际上最终得到了其中一种情况的反例和其他情况的消极前景。从更积极的方面来看，我们已经发现证明嵌入G*U是一个局部开放映射足以得到（JP）的期望解。这个方向已经被发现需要深化我们的前仿射/后仿射理论，例如对我们的拓扑结构的回顾和在我们的范畴中对ettale映射的新定义。我们的研究在这方面取得了很大进展。

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.