Additive preservers of permanent rank

Alexander Guterman; Bojan Kuzma; Leonid Ovchinnikov

Additive preservers of permanent rank

Rings and Algebras 2026-07-24 v1 Combinatorics

Authors: Alexander Guterman , Bojan Kuzma , Leonid Ovchinnikov

Abstract

The permanent rank of a matrix $A$ is the size of the maximal square submatrix in $A$ which has a nonzero permanent. In this paper we characterize additive transformations $\Phi$ which preserve matrices of per-rank-one. Under an additional assumption that $\Phi$ is surjective or that it preserves per-rank-one in both directions we prove that $\Phi$ is a composition of a multiplication with diagonal matrices and permutation matrices from both sides, transposition, and injective endomorphism of the base field.

Keywords

additive preservers permanent rank rank-one permanent linear transformations matrix theory

Cite

@article{arxiv.2607.28664,
  title  = {Additive preservers of permanent rank},
  author = {Alexander Guterman and Bojan Kuzma and Leonid Ovchinnikov},
  journal= {arXiv preprint arXiv:2607.28664},
  year   = {2026}
}