发表机构
Clayton State University（克莱顿州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究除环上满足特定逆函数恒等式的加法映射\(f,g\)何时为零。通过有限域分类、素域和中心子域缩放证明消失定理，探讨特征二情况并证明约化定理，明确了不同条件下\(f,g\)的取值及恒等式的相关结论。
AI 中文摘要
设\(D\)为除环，\(n\geq2\)，\(f,g:D\to D\)为加法映射，满足\(f(x)x^{-1}+x^n g(x^{-1})=0\)（\(x\in D^\times\)）。本文研究此逆函数恒等式何时迫使\(f = g = 0\)以及自然消失陈述不成立的情况。正特征下的主要障碍来自弗罗贝尼乌斯齐次性。首先给出精确的有限域分类，若\(q = p^m\)，则解空间的\(\mathbb{F}_q\)维数是满足\(p^i + p^j\equiv n + 1\pmod{q - 1}\)的有序对\((i,j)\)的数量，\(0\leq i,j\leq m - 1\)。接着用素域和中心子域缩放证明任意除环的消失定理。特征零时恒等式总是迫使\(f = g = 0\)。正特征下，相关中心权重不相容时同样给出消失结果。最后研究特征二的情况，虽未完全解决，但证明了一个约化定理，表明非零值\(f(1)=g(1)\)会产生广义多项式恒等式，除一种明确的中心弗罗贝尼乌斯退化可能性外。特别地，对于非中心有限除环，除\(f(1)=g(1)\)是中心且\(n + 1\)是二的幂次这种特殊情况外，所有特征二的解都简化为\(h(1)=0\)的单映射恒等式。对于\(n = 2\)，不存在这种特殊情况，且约化在每个特征二的非交换除环上成立。
英文摘要
Let $D$ be a division ring, let $n\geq 2$, and let $f,g:D\to D$ be additive maps satisfying $f(x)x^{-1}+x^n g(x^{-1})=0$ for every nonzero $x\in D$. We establish general vanishing criteria, give an explicit classification over finite fields, and determine the additive-polynomial solutions over infinite fields of positive characteristic. If $\mathbb{F}_q\subseteq Z(D)$, every additive map $D\to D$ admits a canonical decomposition into $\mathbb{F}_q^\times$-weight components, and the identity pairs precisely the weights $r,s$ satisfying $r+s\equiv n+1\pmod{q-1}$. Consequently, for $q=p^m$, the dimension over $\mathbb{F}_q$ of the solution space is the number of ordered pairs $(i,j)$ with $0\leq i,j<m$ satisfying $p^i+p^j\equiv n+1\pmod{q-1}$. Over an infinite field of characteristic $p>0$, the additive-polynomial solutions are exactly the sums of paired Frobenius terms with $p^i+p^j=n+1$. This classification concerns additive-polynomial maps and does not assert a classification of arbitrary additive maps. Prime-field dilation gives complete vanishing in characteristic zero and, in characteristic $p>0$, whenever $p-1$ does not divide $n-1$. In characteristic two, we prove complete vanishing for $n=2$ on every noncommutative division ring. More generally, when $[D:Z(D)]=\infty$, every solution vanishes if the center is infinite. If the center is the finite field $\mathbb{F}_q$, the same conclusion holds for $2\leq n\leq q-1$. The remaining cases are stated explicitly.
CommentsNow 23 pages. Some typos have been corrected, and Remark 3.14 has been added. To appear in the Journal of Algebra Combinatorics Discrete Structures and Applications
