交换群间函数的最大可能有限度
The Largest Possible Finite Degree of Functions between Commutative Groups
摘要：
我们考虑交换群和它们的函数度之间的映射。这些度是基于一个简单的想法定义的-如果采取离散导数，则函数度应该减小。我们表明，有限的功能度的地图正是地图，可以写为polyfracts，作为多项式在几个变量，但与二项式函数的地方的权力。此外，polyfract的程度与其功能程度一致。我们用它来确定两个给定的有限交换群之间的映射所能具有的最大可能的有限函数度。这也产生了解决艾兴格和穆斯堡尔的问题找到幂零度的增广理想的群环$Z_{p^β}[Z_{p^{α_1}}\times Z_{p^{α_2}}\times\dots\times Z_{p^{α_n}}]$。对基本事实的证明也作了一些推广和简化。
英文摘要：
We consider maps between commutative groups and their functional degrees. These degrees are defined based on a simple idea -- the functional degree should decrease if a discrete derivative is taken. We show that the maps of finite functional degree are precisely the maps that can be written as polyfracts, as polynomials in several variables but with binomial functions in the place of powers. Moreover, the degree of a polyfract coincides with its functional degree. We use this to determine the largest possible finite functional degree that the maps between two given finite commutative groups can have. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency degree of the augmentation ideal of the group ring $Z_{p^β}[Z_{p^{α_1}}\times Z_{p^{α_2}}\times\dots\times Z_{p^{α_n}}]$. Some generalizations and simplifications of proofs to underlying facts are presented, too.
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