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 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 $r+s\equiv n+1(\mathrm{mod}q-1)$. Consequently, for $q=p^m$, the ${\mathbb{F}}_q$-vector space ${\mathcal{S}}_n({\mathbb{F}}_q)$ of solutions over ${\mathbb{F}}_q$ satisfies ${\dim }_{{\mathbb{F}}_q}{\mathcal{S}}_n({\mathbb{F}}_q)=\left|\left\{(i,j):0\le i,j<m,p^i+p^j\equiv n+1(\mathrm{mod}q-1)\right\}\right|$. Over an infinite field of characteristic $p$, the additive-polynomial solutions are exactly the sums of paired Frobenius terms satisfying $p^i+p^j=n+1$. Prime-field dilation gives complete vanishing in characteristic zero and, in characteristic $p>0$, 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 $2\le n\le q-1$. The remaining finite-center and centrally finite cases are isolated explicitly.

Now 20 pages. The noncommutative case for (n=2) is completely resolved, and several new vanishing theorems in characteristic two have been added. Minor typos have also been corrected. Comments are welcome!