Higher-power inverse functional identities and Frobenius collision obstructions

Abstract

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 all $x\in D^\times$. We establish general vanishing criteria and classify the Frobenius-type obstructions over fields. 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 $\mathbb{F}_q$-vector space $\mathcal{S}_n(\mathbb{F}_q)$ of solutions over $\mathbb{F}_q$ satisfies $\dim_{\mathbb{F}_q}\mathcal{S}_n(\mathbb{F}_q)=\left|\left\{(i,j):0\leq i,j 0$, whenever $(p-1)\nmid(n-1)$. In characteristic two, a generalized-polynomial reduction and an inverse-free identity yield complete vanishing for $n=2$ on every noncommutative division ring. More generally, if $[D:Z(D)]=\infty$, every solution vanishes when the center is infinite. If $Z(D)=\mathbb{F}_q$ is finite, a graded refinement proves the same conclusion for $2\leq n\leq q-1$. The remaining finite-center and centrally finite cases are isolated explicitly.