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- W1519272146 abstract "Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, cdots, f_n$ are elements in a polynomial ring $k[X_1, cdots, X_n]$ over a field $k$ of characteristic zero such that $ det(partial f_i/ partial X_j) $ is a nonzero constant, then $k[f_1, cdots, f_n] = k[X_1, cdots, X_n]$. Practically, what we deal with is the generalized one, noindent The Generalized Jacobian Conjecture$(GJC)$ :{it Let $S hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({sl i.e.,} ${rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^times cap S = S^times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly." @default.
- W1519272146 created "2016-06-24" @default.
- W1519272146 creator A5028886107 @default.
- W1519272146 date "2007-06-08" @default.
- W1519272146 modified "2023-09-27" @default.
- W1519272146 title "Unramified Extensions of an affine regular UFD over a Field of Characteristic zero" @default.
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