The Jacobian Problem concerning Polynomial mappings

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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.