Higher-power inverse functional identities and Frobenius collision obstructions
Abstract
Let D be a division ring, let n≥ 2, and let f,g:D→ D be additive maps satisfying f(x)x−1+xn g(x−1)=0 for every nonzero x∈ 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 𝔽q⊆ Z(D), every additive map D→ D admits a canonical decomposition into 𝔽q×-weight components, and the identity pairs precisely the weights r,s satisfying r+s≡ n+1pmodq−1. Consequently, for q=pm, the dimension over 𝔽q of the solution space is the number of ordered pairs (i,j) with 0≤ i,j<m satisfying pi+pj≡ n+1pmodq−1. Over an infinite field of characteristic p>0, the additive-polynomial solutions are exactly the sums of paired Frobenius terms with pi+pj=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)]=∞, every solution vanishes if the center is infinite. If the center is the finite field 𝔽q, the same conclusion holds for 2≤ n≤ q−1. The remaining cases are stated explicitly.
Related
- Bethe Cycles, Frobenius Identities, and Quantum/Classical Duality
- Inverse trigonometric and hyperbolic identities,series and products for Lucas sequences
- The inverse problem of time-dependent density functional theory on the torus
- Frobenius and quasi-Frobenius left Hopf algebroids
- Frobenius splitting for Fano threefolds
BibTeX
@misc{aliabadi2026higherpower,
title = {{Higher-power inverse functional identities and Frobenius collision obstructions}},
author = {Mohsen Aliabadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.09669},
eprint = {2607.09669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09669},
}
RIS
TY - UNPB
TI - Higher-power inverse functional identities and Frobenius collision obstructions
AU - Mohsen Aliabadi
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.09669
UR - https://arxiv.org/abs/2607.09669
AB - Let D be a division ring, let n≥ 2, and let f,g:D→ D be additive maps satisfying f(x)x^−1+x^n g(x^−1)=0 for every nonzero x∈ 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 𝔽_q⊆ Z(D), every additive map D→ D admits a canonical decomposition into 𝔽_q^×-weight components, and the identity pairs precisely the weights r,s satisfying r+s≡ n+1pmodq−1. Consequently, for q=p^m, the dimension over 𝔽_q of the solution space is the number of ordered pairs (i,j) with 0≤ i,j<m satisfying p^i+p^j≡ n+1pmodq−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)]=∞, every solution vanishes if the center is infinite. If the center is the finite field 𝔽_q, the same conclusion holds for 2≤ n≤ q−1. The remaining cases are stated explicitly.
ER -