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摘要翻译：
我们的目标是解决一个衰落问题，即雅可比猜想$(JC_n)$~:如果$f_1,\cdots,f_n$是特征为零的域$k$上的多项式环$k[X_1,\cdots,X_n]$中的元素，使得$\det(\partial f_i/\partial X_j)$是非零常数，则$k[f_1,\cdots,f_n]=k[X_1,\cdots,X_n]$。实际上，我们研究的是一个广义的Jacobian猜想(GJC)$:它使$S\hookrightarrow T$是Noetherian域的一个未分支同态。假定$S$是一个简单连通的UFD（{\SL,即}${\RM Spec}(S)$是简单连通的，$S$是唯一的因子分解域），并且$T^\times\cap=S^\times$。那么$T=S$。}此外，为了讨论的一致性，我们提出了一些严肃（或愚蠢）的问题，并对某些优秀数学家发表的论文中出现的例子提出了一些评论（尽管我们不愿意处理它们）。然而，这些例子的存在将与我们上面的主要结果相反，因此我们不得不在附录B中对它们关于它们各自的（所谓的）反例子的存在的论点提出争议。我们的结论是，它们不是完全的反例，这是明确地显示出来的。
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英文标题：
《Generalized Jacobian Conjectures -- A purely Algebraic Approach》
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作者：
Susumu Oda
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最新提交年份：
2020
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分类信息：
一级分类：Mathematics 数学
二级分类：Commutative Algebra 交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
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.
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PDF链接：
https://arxiv.org/pdf/0706.1138
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