Higher-power inverse functional identities and Frobenius collision obstructions

Abstract

Let $D$ be a division ring, let $n\ge 2$, 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 $r+s\equiv n+1(\mathrm{mod}q-1)$. Consequently, for $q=p^m$, the dimension over ${\mathbb{F}}_q$ of the solution space is the number of ordered pairs $(i,j)$ with $0\le i,j<m$ satisfying $p^i+p^j\equiv n+1(\mathrm{mod}q-1)$. 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 $2\le n\le q-1$. The remaining cases are stated explicitly.

arXiv commentNow 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