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 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 $0\leq i,j 0$, 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.