234 research outputs found
Generalized Jacobian Conjectures -- A purely Algebraic Approach
Our goal is to settle a fading problem, the Jacobian Conjecture ~:
If are elements in a polynomial ring
over a field of characteristic zero such that is a nonzero constant, then .
Practically, what we deal with is the generalized one,
\noindent
The Generalized Jacobian Conjecture :{\it Let be
an unramified homomorphism of Noetherian domains. Assume that is a simply
connected UFD ({\sl i.e.,} is simply connected and is a
unique factorization domain) and that . Then .}
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.Comment: I do the previous manuscript cleaning and add three figures, and add
Appendix
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